# 4.5: Specific Heat Calculations

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Water has a high capacity for absorbing heat. In a car radiator, it serves to keep the engine cooler than it would otherwise run. As the water circulates through the engine, it absorbs heat from the engine block. When it passes through the radiator, the cooling fan and the exposure to the outside environment allow the water to cool somewhat before it makes another passage through the engine.

## Specific Heat Calculations

The specific heat of a substance can be used to calculate the temperature change that a given substance will undergo when it is either heated or cooled. The equation that relates heat \(\left( q \right)\) to specific heat \(\left( c_p \right)\), mass \(\left( m \right)\), and temperature change \(\left( \Delta T \right)\) is shown below.

\[q = c_p \times m \times \Delta T\]

The heat that is either absorbed or released is measured in joules. The mass is measured in grams. The change in temperature is given by \(\Delta T = T_f - T_i\), where \(T_f\) is the final temperature and \(T_i\) is the initial temperature.

Example \(\PageIndex{1}\)

A \(15.0 \: \text{g}\) piece of cadmium metal absorbs \(134 \: \text{J}\) of heat while rising from \(24.0^\text{o} \text{C}\) to \(62.7^\text{o} \text{C}\). Calculate the specific heat of cadmium.

**Solution**

*Step 1: List the known quantities and plan the problem.*

__Known__

- Heat \(= q = 134 \: \text{J}\)
- Mass \(= m = 15.0 \: \text{g}\)
- \(\Delta T = 62.7^\text{o} \text{C} - 24.0^\text{o} \text{C} = 38.7^\text{o} \text{C}\)

__Unknown__

- \(c_p\) of cadmium \(= ? \: \text{J/g}^\text{o} \text{C}\)

The specific heat equation can be rearranged to solve for the specific heat.

*Step 2: Solve.*

\[c_p = \dfrac{q}{m \times \Delta T} = \dfrac{134 \: \text{J}}{15.0 \: \text{g} \times 38.7^\text{o} \text{C}} = 0.231 \: \text{J/g}^\text{o} \text{C}\]

*Step 3: Think about your result.*

The specific heat of cadmium, a metal, is fairly close to the specific heats of other metals. The result has three significant figures.

Since most specific heats are known, they can be used to determine the final temperature attained by a substance when it is either heated or cooled. Suppose that \(60.0 \: \text{g}\) of water at \(23.52^\text{o} \text{C}\) was cooled by the removal of \(813 \: \text{J}\) of heat. The change in temperature can be calculated using the specific heat equation:

\[\Delta T = \dfrac{q}{c_p \times m} = \dfrac{813 \: \text{J}}{4.18 \: \text{J/g}^\text{o} \text{C} \times 60.0 \: \text{g}} = 3.24^\text{o} \text{C}\]

Since the water was being cooled, the temperature decreases. The final temperature is:

\[T_f = 23.52^\text{o} \text{C} - 3.24^\text{o} \text{C} = 20.28^\text{o} \text{C}\]

## Summary

- The specific heat of a substance can be used to calculate the temperature change of the substance when it is heated or cooled.
- Specific heat calculations are illustrated.

## Contributors and Attributions

CK-12 Foundation by Sharon Bewick, Richard Parsons, Therese Forsythe, Shonna Robinson, and Jean Dupon.