Molar heat capacity is a fundamental property that quantifies the thermal energy necessary to raise the temperature of a substance by a specific amount. The concept is useful for understanding how different materials absorb or store heat, which has applications across various scientific fields. Calculating this value involves measuring the energy transfer and the resulting temperature change in a precisely measured quantity. This article provides a guide to understanding and calculating molar heat capacity using the foundational principles of thermodynamics.
Defining Key Concepts and Variables
Molar heat capacity is defined as the heat energy required to increase the temperature of one mole of a substance by one unit, typically one Kelvin (K) or one degree Celsius (°C). It is an intensive property, meaning the calculated value does not depend on the size or amount of the sample being measured. This makes it a characteristic property of the substance itself.
Molar heat capacity is distinct from specific heat capacity, which is the energy required to raise the temperature of one gram of a substance by one degree. The difference lies in the quantity used: specific heat is normalized per unit of mass, while molar heat capacity is normalized per mole. The standard unit for molar heat capacity is Joules per mole per Kelvin (\(\text{J/mol}\cdot\text{K}\)).
The calculation uses three primary variables, with molar heat capacity represented by the symbol \(C\). \(Q\) is the amount of heat energy transferred into or out of the substance, measured in Joules (J), and \(n\) is the amount of the substance in moles (mol). \(\Delta T\) represents the change in temperature, calculated as the final minus the initial temperature, and is measured in Kelvin or Celsius. Since both scales use the same size degree increments, the numerical value for a temperature change is identical in both units.
Step-by-Step Calculation Using the Core Formula
The fundamental thermodynamic relationship connecting heat energy, amount of substance, and temperature change is \(Q = nC\Delta T\). This equation states that the heat absorbed (\(Q\)) is the product of the amount of substance (\(n\)), the molar heat capacity (\(C\)), and the resulting temperature change (\(\Delta T\)). To solve for \(C\), the formula is algebraically rearranged to isolate it, yielding the working equation: \(C = Q / (n\Delta T)\).
The calculation process requires determining three values. First, the amount of heat energy (\(Q\)) added to the substance is measured, typically using a calorimeter. Second, the amount of the substance in moles (\(n\)) must be found, often by dividing the measured mass of the sample by its known molar mass. Third, the change in temperature (\(\Delta T\)) is measured as the difference between the final and initial temperatures.
For example, if 100 Joules of heat (\(Q\)) are added to 0.1 moles (\(n\)) of a substance, and the temperature increases by 5 Kelvin (\(\Delta T\)), the calculation is straightforward. Substituting these values into the rearranged formula gives \(C = 100 \text{ J} / (0.1 \text{ mol} \times 5 \text{ K})\). Completing the arithmetic yields a molar heat capacity of \(200 \text{ J/mol}\cdot\text{K}\).
Differentiating Molar Heat Capacity at Constant Pressure and Volume
The molar heat capacity of a substance, particularly a gas, is not a single fixed value but depends on the conditions under which the heat is applied. Thermodynamics defines two distinct values: the molar heat capacity at constant pressure (\(C_p\)) and the molar heat capacity at constant volume (\(C_v\)). These values differ because of the work done by or on the substance during the heating process.
When heat is added while the volume is held constant, all the energy goes directly into increasing the internal energy and the temperature, which is the definition of \(C_v\). If the heat is added while the pressure is held constant, the substance is free to expand. This expansion means some of the energy supplied is used to perform mechanical work against the surroundings, such as pushing back the atmosphere.
Because some energy is diverted to expansion work in the constant pressure case, a greater amount of heat must be added to achieve the same temperature rise compared to the constant volume case. This means that \(C_p\) is always greater than \(C_v\). For ideal gases, the relationship between these two values is precisely defined by Mayer’s relation: \(C_p – C_v = R\), where \(R\) is the universal gas constant.