High School Mechanics
EQUATIONS
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INFORMATION
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$$ \large \bar{a} = \frac{\Delta \vec{v}}{\Delta t} $$
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\( {\bar{a}} = average \hspace{4 pt} acceleration \hspace{4 pt} (m \hspace{2 pt} s^{-2}) \\ {\Delta} \vec{v} = change \hspace{4 pt} in \hspace{4 pt} velocity \hspace{4 pt} (m \hspace{2 pt} s^{-1}) \\ {\Delta t} = duration \hspace{4 pt} (s) \) |
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$$ \large
\vec{v_f} = \vec{v_i} + \vec{a}t \\ $$ $$ \large
{v_f}^2 = {v_i}^2 + 2 \vec{a} \cdot (\vec{r} - \vec{r_0}) \\ $$ $$ \large
\vec{r} = \vec{r_0} + \vec{v_i} t + \frac{1}{2}\vec{a} t^2 \\ $$ $$ \large
\vec{r} = \vec{r}_0 + \frac{1}{2} (\vec{v_f} + \vec{v_i}) t \\ $$ $$ \large
\vec{r} = \vec{r}_0 + \vec{v}_ft - \frac{1}{2} \vec{a} t^2 $$
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\(\\ \vec{v_f} = final \hspace{4 pt} velocity \hspace{4 pt} (m \hspace{2 pt} s^{-1})\\ \vec{v_i} = initial \hspace{4 pt} velocity \hspace{4 pt} (m \hspace{2 pt} s^{-1})\\ \vec{a} = acceleration \hspace{4 pt} (m \hspace{2 pt} s^{-2})\\ \textbf{t} = time \hspace{4 pt} (s)\\ \vec{r} = final \hspace{4 pt} position \hspace{4 pt} (m) \\ \vec{r_0} = initial \hspace{4 pt} position \hspace{4 pt} (m)\) |
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$$ \large \sum {\vec{p}_{init, \hspace{2 pt} i}} = \sum{\vec{p}_{final, \hspace{2 pt} i}} $$
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\(\\ \vec{p}_{init, \hspace{2 pt} i} = initital \hspace{4 pt} momentum \hspace{4 pt} contribution \\ \hspace{23 pt} (kg \hspace{2 pt} m \hspace{2 pt} s^{-1}) \\ \vec{p}_{final, \hspace{2 pt} i} = final \hspace{4 pt} momentum \hspace{4 pt} contribution \\ \hspace{30 pt} (kg \hspace{2 pt} m \hspace{2 pt} s^{-1}) \\\) |
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$$ \large J = F_{net} \hspace{2 pt} t = \Delta ps $$
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\(\\ \textbf{J} = impulse \hspace{4 pt} (N \hspace{2 pt} s) \\ {F_{net}} = net \hspace{4 pt} force \hspace{4 pt} (N) \\ \textbf{t} = time \hspace{4 pt} of \hspace{4 pt} collision \hspace{4 pt} (s) \\ {\Delta p} = change \hspace{4 pt} in \hspace{4 pt} momentum \hspace{4 pt} (N \hspace{2 pt} s)\) |
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$$ \large U = mgh $$
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\( \textbf{U} = potential \hspace{4 pt} energy \hspace{4 pt} (J) \\ \textbf{m} = mass \hspace{4 pt} (kg) \\ \textbf{g} = acceleration \hspace{4 pt} due \hspace{4 pt} to \hspace{4 pt} gravity \\ \hspace{8 pt} (9.81 \hspace{4 pt} m \hspace{2 pt} s^{-2}) \\ \textbf{h} = height \hspace{4 pt} (m) \) |
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$$ \large U = \frac{1}{2} k x^2 $$
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\(\\ \textbf{U} = potential \hspace{4 pt} energy \hspace{4 pt} (J) \\ \textbf{m} = mass \hspace{4 pt} (kg) \\ \vec{g} = acceleration \hspace{4 pt} due \hspace{4 pt} to \hspace{4 pt} gravity \\ \hspace{4 pt} (9.81 \hspace{2 pt} m \hspace{2 pt} s^{-2}) \\ \textbf{h} = height \hspace{4 pt} (m)\) |
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$$ \large {{ME}}_{total} = K + U $$
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\(\\ {{{ME}}_{total}} = total \hspace{4 pt} mechanical \hspace{4 pt} energy \hspace{4 pt} (J) \\ \textbf{K} = total \hspace{4 pt} kinetic \hspace{4 pt} energy \hspace{4 pt} (J) \\ \textbf{U} = total \hspace{4 pt} potential \hspace{4 pt} energy \hspace{4 pt} (J)\) |
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$$ \large
F = G \left( \frac{m_1 m_2}{r^2} \right ) $$
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\( \textbf{F} = gravitational \hspace{4 pt} force \hspace{4 pt} between \\ \hspace{6 pt} masses \hspace{4 pt} (N)\\ \textbf{G} = gravitational \hspace{4 pt} constant \\ \hspace{7 pt} (6.67384 x 10^{-11} \hspace{2 pt} m^3 \hspace{2 pt} kg^{-1} \hspace{2 pt} s^{-2}) \\ {m_\#} = mass \hspace{4 pt} of \hspace{4 pt} object \hspace{4 pt} \# \hspace{4 pt} (kg)\\ \textbf{r} = distance \hspace{4 pt} between \hspace{4 pt} objects \hspace{4 pt} (m)\) |
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$$ \large \vec{F_N} = m \vec{g} \hspace{2 pt} cos \hspace{2 pt} \theta $$
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\(\\ \vec{F_N} = normal \hspace{4 pt} force, \hspace{4 pt} perpendicular \\ \hspace{11 pt} to \hspace{4 pt} the \hspace{4 pt} surface \hspace{4 pt} (N) \\ \textbf{m} = mass \hspace{4 pt} (kg) \\ \vec{g} = acceleration \hspace{4 pt} due \hspace{4 pt} to \hspace{4 pt} gravity, \\ \hspace{4 pt}(9.81 \hspace{2 pt} m \hspace{2 pt} s^{-2}) \\ {\theta} = angle \hspace{4 pt} of \hspace{4 pt} the \hspace{4 pt} inclined \\ \hspace{4 pt} measured \hspace{4 pt} from \hspace{4 pt} the \hspace{4 pt} horizontal \hspace{4 pt} (deg)\) |
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$$ \large \vec{F_f} \leq \mu \vec{F_N} $$
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\(\\ \vec{F_f} = force \hspace{4 pt} due \hspace{4 pt} to \hspace{4 pt} friction \hspace{4 pt} exerted \\ \hspace{11 pt} by \hspace{4 pt} each \hspace{4 pt} object\hspace{4 pt} on \hspace{4 pt} the \hspace{4 pt} other \\ \hspace{11 pt} in \hspace{4 pt} the \hspace{4 pt} direction \hspace{4 pt} that \hspace{4 pt} opposes \\ \hspace{11 pt} the \hspace{4 pt} direction \hspace{4 pt} of \hspace{4 pt}the \hspace{4 pt} net \\ \hspace{11 pt} applied \hspace{4 pt} force \hspace{4 pt}(N) \\ {\mu} = coefficient \hspace{4 pt} of \hspace{4 pt} friction \\ \vec{F_N} = normal \hspace{4 pt} force, \hspace{4 pt} perpendicular \\ \hspace{11 pt} to \hspace{4 pt} the \hspace{4 pt} force \hspace{4 pt} of \hspace{4 pt} friction \hspace{4 pt} (N)\) |
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$$ \large v_T = \frac{2 \pi r}{T} $$
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\(\\ {v_T} = tangential \hspace{4 pt} velocity \hspace{4 pt} (m \hspace{2 pt} s^{-1}) \\ \textbf{r} = radius \hspace{4 pt} of \hspace{4 pt} circle \hspace{4 pt} (m)\\ \textbf{T} = period \hspace{4 pt} of \hspace{4 pt} rotation \hspace{4 pt} (s)\) |
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$$ \large \vec{a_c} = \frac{\vec{v}^{\hspace{2 pt} 2}}{r} $$
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\(\\ \vec{a_c} = centripetal \hspace{4 pt} acceleration, \\ \hspace{7 pt} pointed \hspace{4 pt} towards \hspace{4 pt} the \hspace{4 pt} center \hspace{4 pt} (m \hspace{2 pt} s^{-2}) \\ \vec{v} = tangential \hspace{4 pt} velocity, \hspace{4 pt} perpendicular \\ \hspace{4 pt} to \hspace{4 pt} the \hspace{4 pt} acceleration \hspace{4 pt} vector \hspace{4 pt} (m \hspace{2 pt} s^{-1}) \\ \textbf{r} = radial \hspace{4 pt} distance \hspace{4 pt} (m)\) |
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$$ \large \vec{F_c} = m\vec{a_c} = \frac{m\vec{v}^{\hspace{2 pt} 2}}{r} $$
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\(\\ \vec{F_c} = centripetal \hspace{4 pt} force, \hspace{4 pt} pointed \\ \hspace{9 pt} towards \hspace{4 pt} the \hspace{4 pt} center \hspace{4 pt} (N) \\ \textbf{m} = mass \hspace{4 pt} of \hspace{4 pt} object \hspace{4 pt} in \hspace{4 pt} motion \hspace{4 pt} (kg) \\ \vec{a_c} = centripetal \hspace{4 pt} acceleration, \\ \hspace{7 pt} pointed \hspace{4 pt} towards \hspace{4 pt} the \hspace{4 pt} center \hspace{4 pt} (m \hspace{2 pt} s^{-2}) \\ \textbf{r} = radial \hspace{4 pt} distance \hspace{4 pt} (m)\) |
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$$ \large \tau = \left \| \vec{r} \right \| \left \| \vec{F} \right \| \hspace{2 pt} sin \hspace{2 pt} \theta $$
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\(\\ {\tau} = torque \hspace{4 pt} (N \hspace{2 pt} m) \\ \vec{r} = distance \hspace{4 pt} between \hspace{4 pt} where \hspace{4 pt} force \\ \hspace{4 pt} is \hspace{4 pt} applied \hspace{4 pt} and \hspace{4 pt} where \hspace{4 pt} torque \\ \hspace{4 pt} is \hspace{4 pt} measured \hspace{4 pt} (m) \\ \vec{F} = force \hspace{4 pt} applied \hspace{4 pt} (N)\\ {\theta} = angle \hspace{4 pt} between \hspace{4 pt} the \hspace{4 pt} r \hspace{4 pt} and \hspace{4 pt} F \\ \hspace{4 pt} vectors \hspace{4 pt} (deg) \) |
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$$ \large v_{esc} = \sqrt{\frac{2GM}{r}} $$
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\(\\ {v_{esc}} = escape \hspace{4 pt} velocity \hspace{4 pt} (m \hspace{2 pt} s^{-1}) \\ \textbf{G} = gravitational \hspace{4 pt} constant \hspace{4 pt} \\ \hspace{8 pt} (6.67 \hspace{2 pt} x \hspace{2 pt} 10^{-11} \hspace{2 pt} m^3 \hspace{2 pt} kg^{-1} \hspace{2 pt} s^{-2}) \\ \textbf{M} = mass \hspace{4 pt} of \hspace{4 pt} planet \hspace{4 pt} (kg) \\ \textbf{r} = distance \hspace{4 pt} from \hspace{4 pt} center \hspace{4 pt} of \\ \hspace{3 pt} gravity \hspace{4 pt} (m)\) |
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$$ \large T = 2 \pi \sqrt{\frac{L}{g}} $$
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\(\\ \textbf{T} = time \hspace{4 pt} period \hspace{4 pt} of \hspace{4 pt} pendulum \hspace{4 pt} (s) \\ \textbf{L} = length \hspace{4 pt} of \hspace{4 pt} pendulum\\ \textbf{g} = magnitude \hspace{4 pt} of \hspace{4 pt} acceleration \\ \hspace{4 pt} due \hspace{4 pt} to \hspace{4 pt} gravity \hspace{4 pt} (9.81 \hspace{2 pt} m \hspace{2 pt} s^{-2})\) |
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$$ \large f = \frac{v}{\lambda} = \frac{n}{t} $$
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\(\\ \textbf{f} = frequency \hspace{4 pt} (s^{-1}) \\ \textbf{v} = velocity \hspace{4 pt} at \hspace{4 pt} which \hspace{4 pt} wave \\ \hspace{5 pt} propagates \hspace{4 pt} (m \hspace{2 pt} s^{-1}) \\ {\lambda} = wavelength \hspace{4 pt} (m) \\ {n} = number \hspace{4 pt} of \hspace{4 pt} times \hspace{4 pt} an \\ \hspace{5 pt} occurred \\ \textbf{t} = time \hspace{4 pt} of \hspace{4 pt} observation \hspace{4 pt} (s)\) |
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$$ \large f = \left (\frac{c+v_r}{c+v_s} \right ) f_o $$
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\(\\ \textbf{f} = observed \hspace{4 pt} frequency \hspace{4 pt} (s^{-1})\\ \textbf{c} = velocity \hspace{4 pt} of \hspace{4 pt} waves \hspace{4 pt} in \hspace{4 pt} medium \hspace{4 pt} (m \hspace{2 pt} s^{-1}) \\ {v_r} = velocity \hspace{4 pt} of \hspace{4 pt} the \hspace{4 pt} receiver, \\ \hspace{8 pt} positive \hspace{4 pt} if \hspace{4 pt} it's \hspace{4 pt} moving \\ \hspace{8 pt} toward \hspace{4 pt} the \hspace{4 pt} source \hspace{4 pt} and \\ \hspace{8 pt} negative \hspace{4 pt} in \hspace{4 pt} the \hspace{4 pt} other \\ \hspace{8 pt} direction \hspace{4 pt} (m \hspace{2 pt} s^{-1}) \\ {v_s} = velocity \hspace{4 pt} of \hspace{4 pt} the \hspace{4 pt} source, \\ \hspace{8 pt} negative \hspace{4 pt} if \hspace{4 pt} it's \hspace{4 pt} moving \\ \hspace{8 pt} toward \hspace{4 pt} the \hspace{4 pt} receiver \hspace{4 pt} and \\ \hspace{8 pt} positive \hspace{4 pt} in \hspace{4 pt} the \hspace{4 pt} other \\ \hspace{8 pt} direction \hspace{4 pt} (m \hspace{2 pt} s^{-1}) \\ {f_o} = emitted \hspace{4 pt} frequency \hspace{4 pt} (s^{-1})\) |
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$$ \large \omega = \frac{v}{r} $$
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\(\\ {\omega} = angular \hspace{4 pt} frequency \hspace{2 pt} / \hspace{2 pt} rotational \\ \hspace{2 pt}\hspace{4 pt} speed \hspace{4 pt} (rad \hspace{4 pt} s^{-1}) \\ \textbf{v} = tangential \hspace{4 pt} speed \hspace{4 pt} (m \hspace{2 pt} s^{-1})\\ \textbf{r} = radial \hspace{4 pt} distance \hspace{4 pt} (m)\) |
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$$ \large \omega = \sqrt{\frac{k}{m}} $$
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\(\\ {\omega} = angular \hspace{4 pt} frequency \hspace{2 pt} / \hspace{2 pt} natural \\ \hspace{4 pt} frequency \hspace{4 pt} (rad \hspace{4 pt} s^{-1}) \\ \textbf{k} = spring \hspace{4 pt} constant \hspace{4 pt} (N \hspace{2 pt} m^{-1})\\ \textbf{m} = mass \hspace{4 pt} of \hspace{4 pt} object \hspace{4 pt} attached \\ \hspace{9 pt} to \hspace{4 pt} spring \hspace{4 pt} (kg)\) |
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$$ \large f = \frac{Nv}{2d} $$
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\(\\ \textbf{f} = resonant \hspace{4 pt} frequencies \hspace{4 pt} (s^{-1}) \\ {N} = number \hspace{4 pt} of \hspace{4 pt} nodes, \\ \hspace{8 pt} must \hspace{4 pt} be \hspace{4 pt} a \hspace{4 pt} natural \\ \hspace{8 pt} number \hspace{4 pt} \\ \textbf{v} = velocity \hspace{4 pt} of \hspace{4 pt} the \hspace{4 pt} wave \hspace{4 pt} (m \hspace{2 pt} s^{-1}) \\ \textbf{d} = string \hspace{4 pt} length \hspace{4 pt} (m) \) |
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$$ \large \beta = 10 \hspace{2 pt} log_{10} \left(\frac{I}{I_0} \right ) $$
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\(\\ {\beta} = sound \hspace{4 pt} intensity \hspace{4 pt} level \hspace{4 pt} (db) \\ \textbf{I} = sound \hspace{4 pt} intensity \hspace{4 pt} (W \hspace{2 pt} m^{-2})\\ {I_0} =standard \hspace{4 pt} reference \hspace{4 pt} sound \\ \hspace{8 pt} intensity \hspace{4 pt} (10^{-12} W \hspace{2 pt} m^{-2})\) |
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$$ \large n \lambda = 2 d \hspace{2 pt} sin \hspace{2 pt} \theta $$
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\(\\ \textbf{n} = order, \hspace{4 pt} must \hspace{4 pt} be \hspace{4 pt} an \hspace{4 pt} integer\\ {\lambda} = wavelength \hspace{4 pt} of \hspace{4 pt} incident \hspace{4 pt} wave \hspace{4 pt} (m)\\ \textbf{d} = lattice \hspace{4 pt} spacing \hspace{4 pt} (m)\\ {\theta} = angle \hspace{4 pt} between \hspace{4 pt} incident \hspace{4 pt} ray \\ \hspace{4 pt} and \hspace{4 pt} scattering \hspace{4 pt} planes \hspace{4 pt} (deg)\) |
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$$ \large PV = k $$
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\(\\ \textbf{P} = pressure \hspace{4 pt} of \hspace{4 pt} the \hspace{4 pt} system \hspace{4 pt} (N \hspace{2 pt} m^{-2}) \\ \textbf{V} = volume \hspace{4 pt} of \hspace{4 pt} the \hspace{4 pt} gas \hspace{4 pt} (m^3) \\ \textbf{k} = constant \hspace{4 pt} value \hspace{4 pt} (N \hspace{2 pt} m)\) |
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$$ \large \frac{P}{T} = k $$
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\(\\ \textbf{P} = pressure \hspace{4 pt} of \hspace{4 pt} system \hspace{4 pt} (N \hspace{2 pt} m^{-2}) \\ \textbf{T} = temperature \hspace{4 pt} of \hspace{4 pt} the \hspace{4 pt} gas \hspace{4 pt} (K) \\ \textbf{k} = constant \hspace{4 pt} value \hspace{4 pt} (N \hspace{2 pt} m^{-2} \hspace{2 pt} K)\) |
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$$ \large \frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2} = k $$
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\(\\ {P_\#} = pressure \hspace{4 pt} of \hspace{4 pt} system \hspace{4 pt} \# \hspace{4 pt} (N \hspace{2 pt} m^{-2}) \\ {V_\#} = volume \hspace{4 pt} of \hspace{4 pt} gas \hspace{4 pt} \# \hspace{4 pt} (m^3) \\ {T_\#} = temperature \hspace{4 pt} of \hspace{4 pt} gas \hspace{4 pt} \# \hspace{4 pt} (K) \\ \textbf{k} = constant \hspace{4 pt} value \hspace{4 pt} (N \hspace{4 pt} m \hspace{4 pt} K^{-1})\) |
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$$ \large PV = nRT $$
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\(\\ \textbf{P} = pressure \hspace{4 pt} (Pa) \\ \textbf{V} = volume \hspace{4 pt} (m^3) \\ {n} = number \hspace{4 pt} of \hspace{4 pt} moles \hspace{4 pt} of \hspace{4 pt} gas \hspace{4 pt} (mol)\\ \textbf{R} = gas \hspace{4 pt} constant \hspace{4 pt} (8.314 \hspace{2 pt} J \hspace{2 pt} K^{-1} \hspace{4 pt} mol^{-1}) \\ \textbf{T} = temperature \hspace{4 pt} (K) \\ {N} = number \hspace{4 pt} of \hspace{4 pt} particles\\ \textbf{k} = Boltzmann \hspace{4 pt} constant \\ \hspace{5 pt} (1.38 \hspace{2 pt} x 10^{-23} \hspace{2 pt} m^2 \hspace{2 pt} kg \hspace{2 pt} s^{-2} \hspace{2 pt} K^{-1}) \\\) |
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