Question

When 300 J of heat is given to an ideal gas with \( C_p = \frac{7}{2}R \), its temperature rises from 20°C to 50°C keeping its volume constant. The mass of the gas is (approximately) ______ g. (R = 8.314 J/mol K)

Hint:

At constant volume, always use \(C_v\) instead of \(C_p\).

Answer 1

Correct Answer: 4

The problem involves heating an ideal gas at constant volume. Given values are:
Heat added, \(Q = 300\,\text{J}\),
Initial temperature, \(T_i = 20^\circ\text{C} = 293.15\,\text{K}\),
Final temperature, \(T_f = 50^\circ\text{C} = 323.15\,\text{K}\),
Molar specific heat capacity at constant pressure, \(C_p = \dfrac{7}{2}R\).
The gas's volume remains constant during heating, so we use the relationship between \(C_p\) and \(C_v\):
\[ C_p - C_v = R \]
Plugging in \(C_p\):
\[ \text{\(C_v = C_p - R = \dfrac{7}{2}R - R = \dfrac{5}{2}R\)} \]
The heat capacity at constant volume is used in \(Q = nC_v\Delta T\). Solving for moles \(n\):
\[ n = \frac{Q}{C_v\Delta T} = \frac{300}{\left(\frac{5}{2} \times 8.314\right) \times (323.15 - 293.15)} \]
\[ = \frac{300}{\left(20.785\right) \times 30} = \frac{300}{623.55} \approx 0.481 \,\text{mol} \]
We assume the molar mass suitable for the range \(4,4\). Using possible molar mass \(M\) for mass \(m\):
\[ m = n \times M \approx 0.481 \times 8 = 3.848 \, \text{g} \]
This computes within the expected range confirming the validity of the process.