Question

A balloon holding 4.50 moles of argon (Ar) gas absorbs 875 J of thermal energy while doing 137 J of work expanding to a larger volume. (a)Find the change in the balloon's internal energy (in J). (b)Calculate the change in temperature of the gas (in K).

Answer 1

The change in the balloon's internal energy is 738 J, and the change in temperature of the argon gas it contains is 13.2 K, calculated using the first law of thermodynamics and the molar specific heat capacity at constant volume for argon.

Step-by-step explanation:

The student's question involves determining the change in the balloon's internal energy and calculating the change in temperature of the argon gas it contains. This is a classic problem in thermodynamics, a branch of physical chemistry.

Change in Internal Energy

The first law of thermodynamics states that the change in internal energy (ΔU) of a system is equal to the heat (Q) added to the system minus the work (W) done by the system:

ΔU = Q - W

In this scenario, the heat absorbed by the gas is 875 J and the work done by the gas is 137 J. Therefore:

ΔU = 875 J - 137 J = 738 J

Change in Temperature of the Gas

To calculate the change in temperature (ΔT), we need to use the specific heat capacity at constant volume for argon and the number of moles (n). The molar specific heat capacity at constant volume for a monatomic ideal gas like argon is approximately 12.5 J/(mol·K).

Using the formula ΔU = n * Cv * ΔT, where Cv is the molar specific heat capacity at constant volume, we can rearrange to solve for ΔT:

ΔT = ΔU / (n * Cv) = 738 J / (4.50 moles * 12.5 J/(mol·K)) = 13.2 K

Answer 2

(a) Therefore, the change in the balloon's internal energy is
`\(738 \, \text{J}\).`

(b) Therefore, the change in temperature of the gas is approximately
`\(34.5 \, \text{K}\).`

(a)To find the change in internal energy
`(\(\Delta U\))`, you can use the first law of thermodynamics:

`\[\Delta U = Q - W\]`

Where:

-
`\(\Delta U\)` = change in internal energy

- Q = heat added to the system

- W = work done by the system

Given:

`\(Q = 875 \, \text{J}\)` (heat absorbed by the gas)

`\(W = 137 \, \text{J}\)` (work done by the gas)

`\[\Delta U = 875 \, \text{J} - 137 \, \text{J}\]`

`\[\Delta U = 738 \, \text{J}\]`

(b) To find the change in temperature
`(\(\Delta T\))` of the gas, you can use the formula:

`\[Q = nC_v\Delta T\]`

Where:

- n = number of moles of gas

-
`\(C_v\)` = molar specific heat at constant volume

Given:

`\(n = 4.50 \, \text{moles}\)` of argon gas

For monatomic gases like argon,
`\(C_v = (3)/(2)R\), where \(R\)` is the molar gas constant.

`\[\Delta T = (Q)/(nC_v)\]`

`\[\Delta T = \frac{875 \, \text{J}}{4.50 \, \text{mol} * (3)/(2)R}\]`

The change in temperature can be calculated using the molar gas constant,
`\(R = 8.314 \, \text{J/mol}\cdot\text{K}\):`

`\[\Delta T = \frac{875 \, \text{J}}{4.50 \, \text{mol} * (3)/(2) * 8.314 \, \text{J/mol}\cdot\text{K}}\]`

`\[\Delta T \approx 34.5 \, \text{K}\]`