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Relation between Cp and Cv

A balloon con...

Question

A balloon containing an ideal gas is initially kept in an evacuated and insulated room. The balloon ruptures and the gas fills up the entire room. Which one of the following statements is TRUE at the end of above process?

The internal energy of the gas decreases from its initial value, but the enthalpy remains constant.

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The internal energy of the gas increases from its initial value, but the enthalpy remains constant.

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Both internal energy and enthalpy of the gas remain constant.

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Both internal energy and enthalpy of the gas increase

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Solution

The correct option is C Both internal energy and enthalpy of the gas remain constant.
The balloon ruptures in an evacuated and insulated room, the gas fills up the entire room. This case related to free expansion.
i.e., $δ Q = 0,$ $δ W = 0,$
According to first law of thermodynamics,
$δ Q = d U + δ W$
$0 = d U + 0$
$d U = 0$
$U_{2} - U_{1} = 0$
$U_{2} = U_{1}$
$m c_{v} T_{2} = m c_{v} T_{1}$
$m c_{p} T_{2} = m c_{p} T_{1}$
$H_{2} = H_{1}$

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