Heat and Temperature of Water Vapor

Student handout: Heat and Temperature of Water Vapor

In this introduction to heat capacity, students determine a derivative that indicates how much the internal energy changes as the temperature changes when volume is held constant.

Small Group Activity schedule 30 min. build Heat Capacity graph, Student handout for each student, A personal or shared writing space for each student to write/draw/sketch. description Student handout (PDF)
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Thermo Heat Capacity Partial Derivatives

What students learn

Heat capacity at constant volume relates to changes in internal energy, i.e. \({\mathit{\unicode{273}}} Q = dU\).
A conceptual definition of heat capacity at constant volume is the derivative of internal energy with respect to temperature without changing the volume.
Not all derivatives are slopes but they are all ratios of small changes.
Heat capacity at constant volume depends on the value of the volume. optional: Heat capacity at constant pressure is NOT \(\left(dU/dT\right)_p\) - you have to account for work done.

A pressure cooker is an enclosed pot that expels air and traps water vapor, which increases the internal pressure. This in turn raises the boiling point of water and allows food to cook at high temperatures.

Imagine you have a large industrial pressure cooker that holds 1 kg of water vapor. You would like to know how responsive the system is to changes in temperature. To do this, you need to determine a characteristic rate: the ratio of the amount of heat needed to change the temperature a small amount and the change in temperature (i.e., the amount of heat transferred per change in temperature).

The graph provided by your instructor shows internal energy and volume contours plotted on temperature and pressure axes.

Estimate the Temperature-Responsiveness: Use the graph to determine this temperature-responsiveness when the volume is held fixed. The initial state of the system corresponds to the black square. Describe your process.
Reflect on Your Process: Why do you think it matters that you held volume constant in the above estimate?
Explore Dependencies on State Variables: Does the value of your estimate depend on the value of the volume? the temperature?

Keywords

Thermo Heat Capacity Partial Derivatives

Learning Outcomes