Question

If \(C_v\) is the specific heat capacity at constant volume of a gas, then the amount of heat required to increase the temperature of 2 moles of the gas from \(27^\circ\text{C}\) to \(127^\circ\text{C}\) at constant volume is

• A. \(100 C_v\)
• B. \(50 C_v\)
• C. \(500 C_v\)
• D. \(300 C_v\)
• E. \(200 C_v\)

Hint:

Always remember that \(\Delta T\) in degrees Celsius is equal to \(\Delta T\) in Kelvin. You don't need to add 273 to both and then subtract; just subtract the Celsius values directly.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
At constant volume, the heat supplied (\(Q_v\)) goes entirely into increasing the internal energy of the gas.

Step 2: Key Formula or Approach:
The heat required at constant volume is given by:
\[ Q_v = n C_v \Delta T \]
where \(n\) is the number of moles and \(\Delta T\) is the change in temperature.

Step 3: Detailed Explanation:
Given:
Number of moles \(n = 2\).
Initial temperature \(T_1 = 27^\circ\text{C}\).
Final temperature \(T_2 = 127^\circ\text{C}\).
Change in temperature \(\Delta T = T_2 - T_1 = 127 - 27 = 100\text{ K}\) (The magnitude of change is same in Celsius and Kelvin).
Substituting values:
\[ Q_v = 2 \times C_v \times 100 = 200 C_v \]

Step 4: Final Answer:
The amount of heat required is \(200 C_v\).