## Specific Heat and Heat Capacity

#### 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 ($J\bullet K^{-1}$). 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 ($\Delta T$) 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 $\frac{J}{K\bullet mol}$.
• the specific heat capacity, often simply called specific heat, which is the heat capacity per unit mass of a pure substance. This is designated cP and cV and its units are given in $\frac{J}{g\bullet K}$.

## 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:

$q=mc_p\Delta T$

$q=nC_P\Delta T$

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
$\frac{J}{mol\bullet K}$. 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:

$\text{36 grams}\times \frac{\text{1 mol }H_2O}{\text{18 g}}=\text{2.0 mol }H_2O$

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

$q=nC_P\Delta T$

$q=(2.0\;\text{mol})\left(75.2\;\frac{J}{mol\bullet K}\right)(10\;K)$

$q=1504\;J$