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A canister filled with 1.3 mol of single-atom helium gas has a temperature of 315 K. What is the approximate total internal energy of the gas?

(Recall that the equation for kinetic energy due to translation in a gas is: [tex]\frac{3}{2} n R T[/tex]; the equation for kinetic energy due to rotation of a molecule in a gas is: [tex]n R T[/tex]; and [tex]R=8.31 J/(mol \cdot K)[/tex].)

A. 5100 J
B. 4300 J
C. 1200 J
D. 9500 J


Sagot :

To determine the total internal energy of a canister filled with 1.3 mol of single-atom helium gas at a temperature of 315 K, we can use the formula for the kinetic energy due to translation in an ideal monoatomic gas.

The formula to calculate the internal energy [tex]\( U \)[/tex] is given by:
[tex]\[ U = \frac{3}{2} n R T \][/tex]

where:
- [tex]\( n \)[/tex] is the number of moles of the gas,
- [tex]\( R \)[/tex] is the universal gas constant,
- [tex]\( T \)[/tex] is the temperature in Kelvin.

Let's break it down:

1. Given:
- [tex]\( n = 1.3 \)[/tex] moles
- [tex]\( R = 8.31 \)[/tex] J/(mol·K)
- [tex]\( T = 315 \)[/tex] K

2. Substituting the given values into the formula:
[tex]\[ U = \frac{3}{2} \cdot 1.3 \cdot 8.31 \cdot 315 \][/tex]

3. Simplifying step-by-step:
- First, calculate [tex]\( \frac{3}{2} \)[/tex]:
[tex]\[ \frac{3}{2} = 1.5 \][/tex]

- Next, multiply the number of moles by the gas constant:
[tex]\[ 1.3 \cdot 8.31 = 10.803 \][/tex]

- Then, multiply this result by the temperature:
[tex]\[ 10.803 \cdot 315 = 3402.945 \][/tex]

- Finally, multiply by 1.5:
[tex]\[ 1.5 \cdot 3402.945 = 5104.4175 \][/tex]

So, the approximate total internal energy of the gas is:
[tex]\[ U \approx 5104.4175 \][/tex] J

Given the provided options, the correct answer is closest to:

A. 5100 J