#### Learning Objective

- Calculate the change in temperature of a substance given its heat capacity and the energy used to heat it

#### Key Points

- Heat capacity is the ratio of the amount of heat energy transferred to an object to the resulting increase in its temperature.
- Molar heat capacity is a measure of the amount of heat necessary to raise the temperature of one mole of a pure substance by one degree K.
- Specific heat capacity is a measure of the amount of heat necessary to raise the temperature of one gram of a pure substance by one degree K.

#### Terms

- heat capacityThe capability of a substance to absorb heat energy; the amount of heat required to raise the temperature of one mole or gram of a substance by one degree Celsius without any change of phase.
- specific heat capacityThe amount of heat that must be added or removed from a unit mass of a substance to change its temperature by one Kelvin.

## Heat Capacity

Heat capacity is an intrinsic physical property of a substance that measures the amount of heat required to change that substance’s temperature by a given amount. In the International System of Units (SI), heat capacity is expressed in units of joules per kelvin ([latex]J\bullet K^{-1}[/latex]). Heat capacity is an extensive property, meaning that it is dependent upon the size/mass of the sample. For instance, a sample containing twice the amount of substance as another sample would require twice the amount of heat energy (Q) to achieve the same change in temperature ([latex]\Delta T[/latex]) as that required to change the temperature of the first sample.

## Molar and Specific Heat Capacities

There are two derived quantities that specify heat capacity as an intensive property (i.e., independent of the size of a sample) of a substance. They are:

- the molar heat capacity, which is the heat capacity
*per mole*of a pure substance. Molar heat capacity is often designated*CP*, to denote heat capacity under constant pressure conditions, as well as*CV*, to denote heat capacity under constant volume conditions. Units of molar heat capacity are [latex]\frac{J}{K\bullet mol}[/latex]. - the specific heat capacity, often simply called specific heat, which is the heat capacity per unit mass of a pure substance. This is designated
*c*_{P }and*c*and its units are given in [latex]\frac{J}{g\bullet K}[/latex]._{V}

## Heat, Enthalpy, and Temperature

Given the molar heat capacity or the specific heat for a pure substance, it is possible to calculate the amount of heat required to raise/lower that substance’s temperature by a given amount. The following two formulas apply:

[latex]q=mc_p\Delta T[/latex]

[latex]q=nC_P\Delta T[/latex]

In these equations, *m* is the substance’s mass in grams (used when calculating with specific heat), and *n* is the number of moles of substance (used when calculating with molar heat capacity).

## Example

*The molar heat capacity of water, CP, is 75.2 *

[latex]\frac{J}{mol\bullet K}[/latex]*. How much heat is required to raise the temperature of 36 grams of water from 300 to 310 K?*

We are given the molar heat capacity of water, so we need to convert the given mass of water to moles:

[latex]\text{36 grams}\times \frac{\text{1 mol }H_2O}{\text{18 g}}=\text{2.0 mol }H_2O[/latex]

Now we can plug our values into the formula that relates heat and heat capacity:

[latex]q=nC_P\Delta T[/latex]

[latex]q=(2.0\;\text{mol})\left(75.2\;\frac{J}{mol\bullet K}\right)(10\;K)[/latex]

[latex]q=1504\;J[/latex]