How much heat does it take to increase the temperature of 1.80 mol of an ideal gas by 50.0 K near room temperature if the gas is held at constant volume and is (b) monatomic?

Verified step by step guidance

1. Identify the type of process and the specific heat capacity: Since the gas is monatomic and the process is at constant volume, use the specific heat capacity at constant volume for a monatomic ideal gas, which is $\frac{3}{2}R$, where $R$ is the universal gas constant.
2. Calculate the amount of heat required using the formula for heat transfer at constant volume: $Q=n{C}_V\mathrm{\Delta }T$. Here, $n$ is the number of moles of the gas, $C_V$ is the specific heat capacity at constant volume, and $\mathrm{\Delta }T$ is the change in temperature.
3. Substitute the values into the formula: Replace $n$ with 1.80 mol, $C_V$ with $\frac{3}{2}R$, and $\mathrm{\Delta }T$ with 50.0 K.
4. Perform the multiplication to find the expression for $Q$, the heat required.
5. To find the numerical value of $Q$, use the value of the gas constant $R=8.314\text{J/mol}\cdot \text{K}$ and calculate the final result.

Key Concepts

Ideal Gas Law

The Ideal Gas Law relates the pressure, volume, temperature, and number of moles of an ideal gas through the equation PV = nRT. This law is fundamental in understanding the behavior of gases under various conditions, particularly when analyzing how changes in temperature affect other properties of the gas.

Heat Capacity at Constant Volume (Cv)

The heat capacity at constant volume (Cv) is the amount of heat required to raise the temperature of a substance by one degree Celsius (or one Kelvin) while keeping its volume constant. For monatomic ideal gases, Cv is typically calculated as (3/2)R per mole, where R is the universal gas constant, making it essential for calculating heat transfer in this scenario.

First Law of Thermodynamics

The First Law of Thermodynamics states that energy cannot be created or destroyed, only transformed. In the context of heating a gas at constant volume, the heat added to the system (Q) is equal to the change in internal energy (ΔU), which can be calculated using the formula Q = nCvΔT, where n is the number of moles and ΔT is the change in temperature.