Attribution-ShareAlike 4.0 International

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We know the following are true (for perfect gases):

$$\frac{c_p}{c_v} = \gamma$$

$$u_2 - u_1 = c_v(T_2-T_1)$$

$$h_2 - h_1 = c_p(T_2-T_1)$$

So:

$$\begin{align*} h_2 - h_1 &= u_2 - u_1 + (p_2v_2 - p_1v_1) \\ c_p(T_2-T_1) &= c_v(T_2-T_1) + R(T_2-T_1) \\ c_p &= c_v + R \\ \\ c_p &= c_v \gamma \\ c_v + R &= c_v\gamma \\ c_v &= \frac{R}{\gamma - 1} \\ \\ \frac{c_p}{\gamma} &= c_v \\ c_p &= \frac{c_p}{\gamma} + R \\ c_p &= \frac{\gamma}{\gamma -1} R \end{align*}$$

This is because if the change in volume, $\mathrm{d}v = 0$, then the work done, $\mathrm{d}w = 0$ also. So applying the (1st Corollary of the) 1st Law to an isochoric process:

$$\mathrm{d}q + \mathrm{d}w = \mathrm{d}u \rightarrow \mathrm{d}q = \mathrm{d}u$$

since $\mathrm{d}w = 0$.

This is because given that pressure, $p$, is constant, work, $w$, can be expressed as:

$$w = -\int^2_1\! p \,\mathrm{d}v = -p(v_2 - v_1)$$

Applying the (1st corollary of the) 1st law to the closed system:

$$\begin{align*} q + w &= u_2 - u_1 \rightarrow q = u_2 - u_1 + p(v_2 - v_1) \\ q &= u_2 + pv_2 - (u_1 + pv_1) \\ &= h_2 - h_1 = \mathrm{d}h \\ \therefore \mathrm{d}q &= \mathrm{d}h \end{align*}$$

Heating a volume of fluid, $V$, at a constant volume requires specific heat $q_v$ where

$$q_v = u_2 - u_1 \therefore c_v = \frac{q_v}{\Delta T}$$

Heating the same volume of fluid but under constant pressure requires a specific heat $q_p$ where

$$q_p =u_2 - u_1 + p(v_2-v_1) \therefore c_p = \frac{q_p}{\Delta T}$$

Since $p(v_2-v_1) > 0$, $\frac{q_p}{q_v} > 1 \therefore q_p > q_v \therefore c_p > c_v$.

The ratio $\frac{c_p}{c_v} = \gamma$

$$\begin{align*} p_1v_1^n &= p_2v_2^n \\ pv &= RT \rightarrow v = R \frac{T}{p} \\ \frac{p_2}{p_1} &= \left( \frac{v_1}{v_2} \right)^n \\ &= \left(\frac{p_2T_1}{T_2p_1}\right)^n \\ &= \left(\frac{p_2}{p_1}\right)^n \left(\frac{T_1}{T_2}\right)^n \\ \left(\frac{p_2}{p_1}\right)^{1-n} &= \left(\frac{T_1}{T_2}\right)^n \\ \frac{p_2}{p_1} &= \left(\frac{T_1}{T_2}\right)^{\frac{n}{1-n}} \\ &= \left(\frac{T_2}{T_1}\right)^{\frac{n}{n-1}} \\ \end{align*}$$

$$\begin{align*} 0 &= s_2 - s_1 = c_v \ln{\left(\frac{p_2}{p_1}\right)} + c_p \ln{\left( \frac{v_2}{v_1} \right)} \\ 0 &= \ln{\left(\frac{p_2}{p_1}\right)} + \frac{c_p}{c_v} \ln{\left( \frac{v_2}{v_1} \right)} \\ &= \ln{\left(\frac{p_2}{p_1}\right)} + \gamma \ln{\left( \frac{v_2}{v_1} \right)} \\ &= \ln{\left(\frac{p_2}{p_1}\right)} + \ln{\left( \frac{v_2}{v_1} \right)^\gamma} \\ &= \ln{\left[\left(\frac{p_2}{p_1}\right)\left( \frac{v_2}{v_1} \right)^\gamma\right]} \\ e^0 = 1 &= \left(\frac{p_2}{p_1}\right)\left( \frac{v_2}{v_1} \right)^\gamma \\ &\therefore p_2v_2^\gamma = p_1v_1^\gamma \\ \\ pv^\gamma = \text{constant} \end{align*}$$