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 (a) diatomic;

Verified step by step guidance

1. Identify the specific heat capacity at constant volume for a diatomic gas. For diatomic gases near room temperature, the molar specific heat capacity at constant volume, $C_V$, is typically about 5R/2, where R is the universal gas constant (approximately 8.314 J/mol·K).
2. Calculate the total heat required using the formula $Q = nC_V\Delta T$, where $Q$ is the heat added, $n$ is the number of moles of the gas, $C_V$ is the molar specific heat capacity at constant volume, and $\Delta T$ is the change in temperature.
3. Substitute the given values into the formula: $n = 1.80$ mol, $C_V = \frac{5}{2}R$, and $\Delta T = 50.0$ K.
4. Perform the multiplication to find the value of $Q$. Remember to use the value of R as 8.314 J/mol·K.
5. The result from the calculation will give you the amount of heat in joules required to increase the temperature of the gas by 50.0 K at constant volume.

Key Concepts

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 the volume constant. For an ideal gas, Cv depends on the number of degrees of freedom of the gas molecules. Diatomic gases, which have more complex molecular structures than monatomic gases, typically have a higher Cv due to additional rotational and vibrational modes.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in thermodynamics that relates the pressure, volume, temperature, and number of moles of an ideal gas. It is expressed as PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature in Kelvin. This law helps in understanding how gases behave under various conditions and is essential for calculating changes in state variables.

Molar Heat Capacity

Molar heat capacity is the amount of heat required to raise the temperature of one mole of a substance by one degree Celsius (or one Kelvin). For diatomic gases, the molar heat capacity at constant volume (Cv,m) is typically around 5R/2, where R is the ideal gas constant. This concept is crucial for calculating the total heat transfer when the temperature of a gas changes, especially under constant volume conditions.