Question 20.2: Heating a Cylinder of Helium A cylinder contains 3.00 mol of...

Heating a Cylinder of Helium

A cylinder contains 3.00 mol of helium gas at a temperature of 300 K.

(A) If the gas is heated at constant volume, how much energy must be transferred by heat to the gas for its temperature to increase to 500 K?

(B) How much energy must be transferred by heat to the gas at constant pressure to raise the temperature to 500 K?

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(A) Conceptualize Run the process in your mind with the help of the piston–cylinder arrangement in Figure 19.7. Imagine that the piston is clamped in position to maintain the constant volume of the gas.

Categorize We evaluate parameters with equations developed in the preceding discussion, so this example is a substitution problem.

Use Equation 20.23 to find the energy transfer:

Q=nCVΔT( constant volume )Q=n C_V \Delta T \quad (\text { constant volume })     (20.23)

Q1=nCVΔTQ_1=n C_V \Delta T

Substitute the given values:

Q1=(3.00 mol)(12.5 J/molK)(500 K300 K)=7.50×103 J\begin{aligned}Q_1 & =(3.00 \text{ mol})(12.5  J/ \text{mol} \cdot K)(500  K-300  K) \\& =7.50 \times 10^3  J\end{aligned}

(B) Use Equation 20.24 to find the energy transfer:

Q=nCPΔT (constant pressure)Q=n C_P \Delta T \quad \text { (constant pressure)}     (20.24)

Q2=nCPΔTQ_2=n C_P \Delta T

Substitute the given values:

Q2=(3.00 mol)(20.8 J/molK)(500 K300 K)=12.5×103 J\begin{aligned}Q_2 & =(3.00 \text{~mol})(20.8  J/ \text{mol} \cdot K)(500  K-300  K) \\& =12.5 \times 10^3  J\end{aligned}

This value is larger than Q1Q_1 because of the transfer of energy out of the gas by work to raise the piston in the constant pressure process.

19.5 20.2

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