Question 800: Thermodynamics & KTG

Question

A diatomic ideal gas is used in Carnot engine as a working substance. If during the adiabatic expansion part of the cycle, the volume of the gas increases from V to 32 V , the efficiency of the engine is

Accepted Answer

The efficiency of a Carnot engine is given by the formula:
$$ \eta = 1 - \frac{T_C}{T_H} $$
where $ T_C $ is the temperature of the cold reservoir and $ T_H $ is the temperature of the hot reservoir.
For adiabatic processes involving an ideal gas, the relationship between temperature and volume is described by:
$$ TV^{\gamma - 1} = 	ext{constant} $$
where $ \gamma $ (gamma) is the heat capacity ratio $ C_p/C_v $. For a diatomic gas, $ \gamma $ is approximately 1.4.
During an adiabatic expansion:
Volume changes from $ V $ to $ 32V $.
The temperatures at the initial and final volumes are related by:
$$     T_1V^{\gamma-1} = T_2(32V)^{\gamma-1}     $$
Simplifying,
$$     \frac{T_2}{T_1} = \left(\frac{1}{32}ight)^{\gamma-1}     $$
Substitute $ \gamma = 1.4 $:
$$ \frac{T_2}{T_1} = \left(\frac{1}{32}ight)^{0.4} $$
Calculate:
$$ \left( \frac{1}{32} ight)^{0.4} \approx \frac{1}{4} 	ext{ or } T_2 = \frac{T_1}{4} $$
Thus, in terms of Carnot efficiency:
$ T_H = T_1 $
$ T_C = T_2 \approx \frac{T_1}{4} $
So,
$$ \eta = 1 - \frac{T_C}{T_H} = 1 - \frac{\frac{T_1}{4}}{T_1} = 1 - \frac{1}{4} = 0.75 $$
Thus, the efficiency of the Carnot engine is:
Option C: 0.75

Answer

0.25

Answer

0.50

Answer

0.75

Answer

0.90

Thermodynamics & KTG

A diatomic ideal gas is used in Carnot engine as a working substance. If during the adiabatic expansion part of the cycle, the volume of the gas increases from V to 32 V , the efficiency of the engine is

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