Question 149: Thermodynamics & KTG

Question

Two moles of ideal helium gas are in a rubber balloon at 30oC. The balloon is fully expandable
and can be assumed to require no energy in its expansion. The temperature of the gas in the balloon is slowly changed to 35oC. The amount of heat required in raising the temperature is
nearly (take R = 8.31 J/mol.K)

Accepted Answer

Here's a breakdown of how to solve this problem:
Understanding the Concepts
Ideal Gas:
An ideal gas is a theoretical gas that follows the ideal gas law perfectly. The ideal gas law is a relationship between pressure ($$P$$), volume ($$V$$), temperature ($$T$$), and the number of moles ($$n$$) of a gas:
$$PV = nRT$$, where $$R$$ is the ideal gas constant.
Heat Capacity:
Heat capacity is the amount of heat energy required to raise the temperature of a substance by 1 degree Celsius (or 1 Kelvin).  For an ideal gas, we can use the specific heat capacity at constant volume ($$C_v$$) or at constant pressure ($$C_p$$).
Constant Volume Process:
In a constant volume process, the volume of the gas remains constant. The heat required to raise the temperature is given by:
$$Q = nC_v\Delta T$$
Constant Pressure Process:
In a constant pressure process, the pressure of the gas remains constant. The heat required to raise the temperature is given by:
$$Q = nC_p\Delta T$$
Molar Heat Capacities of Helium:
Helium is a monatomic gas. For monatomic gases, we have:
$$C_v = \frac{3}{2}R$$
$$C_p = \frac{5}{2}R$$
Solving the Problem
Identify the Process:
Since the balloon is fully expandable, the pressure remains constant. This is a constant pressure process.
Calculate the Heat:
Use the formula for heat at constant pressure:
$$Q = nC_p\Delta T$$
Substitute the Values:
$$n = 2 	ext{ moles}$$
$$C_p = \frac{5}{2}R = \frac{5}{2}(8.31 	ext{ J/mol.K})$$
$$\Delta T = 35^\circ C - 30^\circ C = 5 K$$
Solve for Q:
$$Q = (2 	ext{ moles}) 	imes \left(\frac{5}{2} 	imes 8.31 	ext{ J/mol.K}ight) 	imes (5 K) $$
$$Q = 207.75 	ext{ J}$$
Answer:
The amount of heat required to raise the temperature of the helium gas is approximately
208 J
.  Therefore, the correct answer is
Option D
.

Answer

62 J

Answer

104 J

Answer

124 J

Answer

208 J

Thermodynamics & KTG

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