When a gas expands adiabatically, its volume is doubled while its absolute temperature is decreased by a factor \(1.32\). Compute the number of degrees of freedom for the gas molecules.

**Solution:**

The number of degrees can be found from the relation $$ f=\frac{2}{\gamma-1}$$

We can find \(\gamma\) from the adiabatic relation,$$ T_2V_2^{\gamma-1}= T_1V_1^{\gamma-1} $$

$$( \frac{V_2}{V_1})^{\gamma-1}=\frac{T_1}{T_2}=1.32$$

$$ 2^{\gamma-1}=1.32$$

where $$ \gamma=1+\frac{log 1.32}{log2}=1.4$$

The number of degrees of freedom $$ f=\frac{2}{1.4-1}=5$$

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