Question

Q18.23 The discussion in Section 18.4 concluded that all ideal monatomic gases have the same heat capacity ${\mathbf{C}}_{\mathbf{v}}$. Does this mean that it takes the same amount of heat to raise the temperature of 1.0gof each one by1.0K?Explain your reasoning.

Short answer

No. It takes different amounts of heat to raise the temperature of 1.0g of all monoatomic ideal gases by1.0K .

Step-by-step solution

Definition of heat capacity and expression

The heat capacity is the quantity of heat required to raise the temperature of 1 mole of a monoatomic gas to 1.0K.

The amount of heat absorbed and the temperature increased is related by $Q=n{C}_v∆T$

where Q is the heat intake, $n$is the number of moles, $C_v$is the heat capacity at constant volume, $∆T$is the change in temperature.

Step 2:  Expression for change in temperature

Expression for change in temperature from the above equation is $∆T=\frac{Q}{n{C}_v}$.

Even if the value of heat capacity, is constant, still change in temperature depends on the heat intake, and the number of moles as well.

For different gases, the number of moles contained in 1g will be different. Thus, the amount of heat required will be more for more number of moles (as $Q\propto n$), and the corresponding rise in temperature will be less, as $∆T\propto \frac{1}{n}$.

Hence, it takes different amount of heat to raise the temperature of 1.0g of each sample of monoatomic gas by1.0K .