Question

At constant volume, 1.0 kJ of heat is transferred to 2 moles of an ideal gas at 1 atm and 298 K. The final temperature of the ideal gas is ____K. (round off to one decimal place)
[Given: R = 8.314 J K^-1 mol^-1]

Answer 1

Correct Answer: 338

To determine the final temperature of the ideal gas after transferring 1.0 kJ of heat, we use the equation for heat transfer at constant volume, also known as the specific heat equation: q = nC_vΔT. Here, q is the heat added, n is the number of moles, C_v is the molar specific heat capacity at constant volume, and ΔT is the change in temperature. We need to find ΔT and then calculate the final temperature using the initial temperature.

First, convert the heat from kJ to J for consistency with the R constant: 
1.0 kJ = 1000 J.

Next, determine the molar specific heat capacity, C_v, using the relationship C_v = (3/2)R for an ideal monatomic gas. Thus, C_v is calculated as:
C_v = (3/2) × 8.314 J K^-1 mol^-1 = 12.471 J K^-1 mol^-1.

We rearrange the specific heat equation to solve for the change in temperature:
ΔT = q / (nC_v).

Substitute the known values into the equation:
ΔT = 1000 J / (2 mol × 12.471 J K^-1 mol^-1)
ΔT = 1000 / 24.942 = 40.1 K (rounded to one decimal place).

Add the change in temperature to the initial temperature to find the final temperature:
T_final = T_initial + ΔT
T_final = 298 K + 40.1 K = 338.1 K.

The calculated final temperature is 338.1 K