Question

Find the heat energy that must be supplied to 14 g of nitrogen at room temperature to raise its temperature by \(48^\circ\text{C}\) at constant pressure. (Molecular weight of nitrogen = 28; \(R\) is the gas constant; \(C_p = \frac{7}{2}R\) for a diatomic gas.)

Hint:

Always identify if the gas is monoatomic or diatomic to verify \(C_p\). Nitrogen (\(N_2\)) is diatomic. Also, remember that for calculating heat energy, the change in temperature in Celsius is identical to the change in Kelvin.

Answer 1

Step 1: Understanding the Concept:
The heat energy supplied at constant pressure is given by \(Q = n C_p \Delta T\), where \(n\) is the number of moles and \(\Delta T\) is the change in temperature.

Step 2: Key Formula or Approach:
1. Number of moles \(n = \frac{\text{Mass}}{\text{Molecular Weight}}\).
2. \(Q = n C_p \Delta T\).

Step 3: Detailed Explanation:
1. Calculate moles (\(n\)):
\(n = \frac{14}{28} = 0.5\) mol.
2. Given parameters:
\(C_p = \frac{7}{2} R\).
\(\Delta T = 48^\circ\)C (Note: a change of \(48^\circ\)C is equal to a change of 48 K).
3. Calculate Heat (\(Q\)):
\[ Q = 0.5 \cdot \left( \frac{7}{2} R \right) \cdot 48 \] \[ Q = \frac{1}{2} \cdot \frac{7}{2} \cdot 48 \cdot R \] \[ Q = \frac{7}{4} \cdot 48 \cdot R = 7 \cdot 12 \cdot R = 84 R \]
Step 4: Final Answer:
The heat energy required is \(84 R\).