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}