高校物理 熱力学 ~ 気体分子の運動

$$
p = \frac{F}{S}
$$

p = \frac{F}{S}

\begin{eqnarray*}
\frac{pV}{T} = \text{const.} \\
\\
\frac{p_1 V_1}{T_1} = \frac{p_2 V_2}{T_2}
\end{eqnarray*}

\frac{pV}{T} = \text{const.}

\frac{p_1 V_1}{T_1} = \frac{p_2 V_2}{T_2}

$$
pV = nRT
$$

pV = nRT

$$
P = \frac{N m \overline{v^2}}{3V}
$$

P = \frac{N m \overline{v^2}}{3V}

\begin{eqnarray*}
\overline{v^2} &=& \frac{v_1^2 + v_2^2 + \cdots + v_N^2}{N} \\
\\
\overline{v_x^2} &=& \frac{v_{1x}^2 + v_{2x}^2 + \cdots + v_{Nx}^2}{N} \\
\\
v_i^2 &=& v_{ix}^2 + v_{iy}^2 + v_{iz}^2 \\
\\
\overline{v^2} &=& \frac{v_1^2 + v_2^2 + \cdots + v_N^2}{N} \\
\\
&=& \frac{1}{N}{(v_{1x}^2 + v_{2x}^2 + \cdots + v_{Nx}^2) + (v_{1y}^2 + v_{2y}^2 + \cdots + v_{Ny}^2) + (v_{1z}^2 + v_{2z}^2 + \cdots + v_{Nz}^2)} \\
\\
&=& \overline{v_x^2} + \overline{v_y^2} + \overline{v_z^2} \\
\\
&=& 3 \overline{v_x^2} \\
\\
\overline{v_x^2} &=& \frac{\overline{v^2}}{3}
\end{eqnarray*}

\overline{v^2} &=& \frac{v_1^2 + v_2^2 + \cdots + v_N^2}{N}

\overline{v_x^2} &=& \frac{v_{1x}^2 + v_{2x}^2 + \cdots + v_{Nx}^2}{N}

v_i^2 &=& v_{ix}^2 + v_{iy}^2 + v_{iz}^2

\overline{v^2} &=& \frac{v_1^2 + v_2^2 + \cdots + v_N^2}{N}

&=& \frac{1}{N}{(v_{1x}^2 + v_{2x}^2 + \cdots + v_{Nx}^2) + (v_{1y}^2 + v_{2y}^2 + \cdots + v_{Ny}^2) + (v_{1z}^2 + v_{2z}^2 + \cdots + v_{Nz}^2)}

&=& \overline{v_x^2} + \overline{v_y^2} + \overline{v_z^2}

&=& 3 \overline{v_x^2}

\overline{v_x^2} &=& \frac{\overline{v^2}}{3}

\begin{eqnarray*}
\overline{\frac{1}{2} mv^2} &=& \frac{1}{N} \left( \frac{1}{2} mv_1^2 + \frac{1}{2} mv_2^2 + \cdots + \frac{1}{2} mv_N^2 \right) = \frac{1}{2} m \overline{v^2} \\
\\
N \cdot \frac{1}{2} m \overline{v^2} &=& n \cdot \frac{3}{2} RT \\
\\
\frac{1}{2} m \overline{v^2} &=& \frac{3}{2} \frac{R}{N_\text{A}} T = \frac{3}{2} kT
\end{eqnarray*}

\overline{\frac{1}{2} mv^2} &=& \frac{1}{N} \left( \frac{1}{2} mv_1^2 + \frac{1}{2} mv_2^2 + \cdots + \frac{1}{2} mv_N^2 \right) = \frac{1}{2} m \overline{v^2}

N \cdot \frac{1}{2} m \overline{v^2} &=& n \cdot \frac{3}{2} RT

\frac{1}{2} m \overline{v^2} &=& \frac{3}{2} \frac{R}{N_\text{A}} T = \frac{3}{2} kT

$$
k = \frac{R}{N_\text{A}} = \frac{8.31 \ \mbox{J/(mol} \cdot \mbox{K)}}{6.02 \times 10^{23} \ \mbox{1/mol}} \fallingdotseq 1.38 \times 10^{-23} \ \mbox{J/K}
$$

$$
\sqrt{\overline{v^2}} = \sqrt{\frac{3RT}{N_\text{A} m}} = \sqrt{\frac{3RT}{M}}
$$

\sqrt{\overline{v^2}} = \sqrt{\frac{3RT}{N_\text{A} m}} = \sqrt{\frac{3RT}{M}}

\begin{eqnarray*}
K &=& \frac{3}{2} nRT \ (\text{単原子分子 monoatomic molecule}) \\
\\
K &=& \frac{3}{2} nRT + nRT = \frac{5}{2} nRT \ (\text{二原子分子 diatomic molecule})
\end{eqnarray*}

K &=& \frac{3}{2} nRT \ (\text{単原子分子 monoatomic molecule})

K &=& \frac{3}{2} nRT + nRT = \frac{5}{2} nRT \ (\text{二原子分子 diatomic molecule})

\begin{eqnarray*}
W &=& pS \cdot L \\
\\
&=& p \cdot SL \\
\\
&=& p \cdot (V_2 – V_2) \\
\\
&=& p \Delta V
\end{eqnarray*}

W &=& pS \cdot L

&=& p \cdot SL

&=& p \cdot (V_2 – V_2)

&=& p \Delta V

$$
Q_V = n C_V \Delta T
$$

Q_V = n C_V \Delta T

$$
Q_p = n C_p \Delta T
$$

Q_p = n C_p \Delta T

\begin{eqnarray*}
\Delta U &=& n C_V \Delta T \\
\\
U &=& n C_V T
\end{eqnarray*}

\Delta U = n C_V \Delta T

U = n C_V T

$$
C_p = C_V +R
$$

C_p = C_V +R

\begin{eqnarray*}
C_V = \frac{3}{2} R \ &,& \ C_p = \frac{5}{2} R \ (\text{単原子分子 monoatomic molecule}) \\
\\
C_V = \frac{5}{2} R \ &,& \ C_p = \frac{7}{2} R \ (\text{二原子分子 diatomic molecule})
\end{eqnarray*}

C_V = \frac{3}{2} R \ &,& \ C_p = \frac{5}{2} R \ (\text{単原子分子 monoatomic molecule})

C_V = \frac{5}{2} R \ &,& \ C_p = \frac{7}{2} R \ (\text{二原子分子 diatomic molecule})

$$
e = \frac{W’}{Q_1} = \frac{Q_1 – Q_2}{Q_1}
$$

e = \frac{W’}{Q_1} = \frac{Q_1 – Q_2}{Q_1}