Question

What fraction of applied heat is converted to work on the surroundings by air expanding isobarically against a piston.

Answer 1

Answer on Question #78383, Physics / Molecular Physics | Thermodynamics

Question. What fraction of applied heat is converted to work on the surroundings by air expanding isobarically against a piston.

Answer. According to the First law of thermodynamics

$$\delta Q = d U + \delta A$$ $$\delta A = p d V = v R d T$$

If $p = \text{const}$ then

$$C_p = \frac{\delta Q}{v d T} \quad \rightarrow \quad \delta Q = C_p v d T$$ $$d U = C_v v d T$$

We have

$$C_p v d T = C_v v d T + \boldsymbol{v} \boldsymbol{R} \boldsymbol{d} \boldsymbol{T} \quad \rightarrow \quad C_p = C_v + \boldsymbol{R} \quad \rightarrow \quad 1 = \frac{C_v}{C_p} + \frac{R}{C_p}$$

For air $C_p = 29.133 \frac{J_{\text{mol} \cdot K}}{\text{mol} \cdot K}$ and C_v = 20.764 \frac{J_{\text{mol} \cdot K}}}{\text{mol} \cdot K}. Finally

$$1 = \frac{C_v}{C_p} + \frac{R}{C_p} = \frac{20.764}{29.133} + \frac{8.31}{29.133} \approx 0.71 + 0.29$$

So, $\sim 29\%$ of applied heat is converted to work.

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