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
