19.11
Heat Capacity: Problem-Solving
The heat capacity of a gas is the amount of heat energy required to raise the temperature of a unit mass of gas by one degree Celsius. It is an import…
Consider a metal cylinder containing 160 grams of oxygen gas with a molar mass of 16 g/mol in a room at 25.0 degrees Celsius. The cylinder is moved and kept outside the room on a hot summer day.
The oxygen gas in the cylinder comes into equilibrium with the surrounding temperature, as 600 J of heat is conducted through the cylinder walls. Ignoring the expansion of the metal cylinder, determine the equilibrium temperature.
To solve the problem, first, identify the known and unknown quantities.
Next, the number of moles can be determined by dividing the mass of the gas by its molar mass.
Recall the molar heat capacity equation at constant volume.
Oxygen is a diatomic gas; by substituting the degree of freedom, the equation of molar heat capacity for an ideal diatomic gas at a constant volume can be obtained.
Lastly, by rearranging the equation and substituting the values, the temperature change is determined to be 2.88 degrees Celsius, and the equilibrium temperature to be 27.88 degrees Celsius.
View the full transcript and gain access to JoVE Core videos
Q1: How do you calculate the number of moles from mass and molar mass?
To find the number of moles, divide the mass of the gas by its molar mass. For example, 160 grams of oxygen gas with a molar mass of 16 g/mol yields 10 moles. This calculation is essential for determining heat capacity and temperature changes in gas problems.
Q2: What is the molar heat capacity equation for diatomic gases at constant volume?
The molar heat capacity at constant volume for an ideal diatomic gas depends on its degree of freedom. Oxygen, a diatomic gas, has five degrees of freedom. Using the kinetic theory of an ideal gas, the molar heat capacity equation uses the gas constant and degree of freedom to calculate heat energy required to raise one mole by one degree Celsius.
Q3: How does molecular structure affect a gas's heat capacity?
A gas's heat capacity depends on its molecular structure and degree of freedom. Monatomic gases like helium have three degrees of freedom, while diatomic gases like oxygen have five. The degree of freedom determines how many directions molecules can move, directly affecting the heat energy required to raise temperature.
Q4: What is the difference between heat capacity at constant volume and constant pressure?
Heat capacity at constant volume (CV) measures heat energy needed to raise one mole of gas by one degree Celsius while volume remains fixed. Heat capacity at constant pressure (CP) measures the same for fixed pressure conditions. Both depend on molecular structure, but CP is always greater than CV due to expansion work.
Q5: How do you find the equilibrium temperature when heat is conducted through a cylinder?
Rearrange the molar heat capacity equation to solve for temperature change using the heat conducted, number of moles, and molar heat capacity. For oxygen receiving 600 J of heat, the temperature change is 2.88 degrees Celsius. Add this to the initial temperature to find the equilibrium temperature of 27.88 degrees Celsius.
Q6: Why is the degree of freedom important in heat capacity calculations?
Degree of freedom represents the number of directions a gas molecule can move in a dynamic system. It directly determines the molar heat capacity value through the equation CV = (d/2)R, where d is degree of freedom and R is the gas constant. Different molecular structures have different degrees of freedom, affecting their heat capacity values.
Q7: What steps should you follow to solve a heat capacity problem?
First, identify known and unknown quantities including mass, molar mass, and heat conducted. Calculate moles by dividing mass by molar mass. Determine the gas type and its degree of freedom. Apply the molar heat capacity equation at constant volume, then rearrange to solve for temperature change and equilibrium temperature.