To raise the temperature of a certain mass of gas by 50°C at a constant pressure, 160 calories of heat is required. When the same mass of gas is cooled by 100°C at constant volume, 240 calories of heat is released. How many degrees of freedom does each molecule of this gas have (assume gas to be ideal)?

We begin by recalling the two calorimetric relations for an ideal gas:

At constant pressure: $$Q_P = n\,C_P\,\Delta T$$

At constant volume: $$Q_V = n\,C_V\,\Delta T$$

Here $$n$$ is the number of moles of the gas, $$C_P$$ and $$C_V$$ are its molar heat capacities at constant pressure and constant volume respectively, and $$\Delta T$$ is the change in temperature expressed in either °C or K (the difference is the same).

For the first experiment we are told that the temperature is raised by $$50^\circ{\rm C}$$ at constant pressure and that the gas absorbs $$160\ {\rm cal}$$ of heat. Substituting these data into the pressure-process formula we get

$$160 = n\,C_P\,(+50)$$

Dividing both sides by $$50$$ we obtain

$$n\,C_P = \frac{160}{50} = 3.2\ {\rm cal\;K^{-1}}.$$

In the second experiment the same mass of gas is cooled by $$100^\circ{\rm C}$$ at constant volume and releases $$240\ {\rm cal}$$ of heat. Heat released is taken as negative, so we write

$$Q_V = -240\ {\rm cal},\quad \Delta T = -100^\circ{\rm C}.$$

Now substituting in the volume-process formula,

$$-240 = n\,C_V\,(-100).$$

Both minus signs cancel, giving

$$n\,C_V = \frac{240}{100} = 2.4\ {\rm cal\;K^{-1}}.$$

An ideal gas obeys the universal relation

$$C_P - C_V = R,$$

so for our sample

$$n\,C_P - n\,C_V = n\,R.$$

We substitute the values already obtained:

$$3.2 - 2.4 = n\,R \;\Longrightarrow\; n\,R = 0.8\ {\rm cal\;K^{-1}}.$$

Next we express the molar heat capacity at constant volume in terms of the number of degrees of freedom $$f$$. For an ideal gas,

$$C_V = \dfrac{f}{2}\,R \quad\Longrightarrow\quad n\,C_V = \dfrac{f}{2}\,n\,R.$$

We know both $$n\,C_V$$ and $$n\,R$$, so we substitute:

$$2.4 = \dfrac{f}{2}\,(0.8).$$

Solving for $$f$$,

$$\dfrac{f}{2} = \dfrac{2.4}{0.8} = 3 \;\Longrightarrow\; f = 6.$$

The gas molecules therefore possess six degrees of freedom, which corresponds to Option B.

Hence, the correct answer is Option B.