## Specific Heat and Heat Capacity

Heat capacity is a measure of the amount of heat energy required to change the temperature of a pure substance by a given amount.

### Learning Objectives

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

### Key Takeaways

#### 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.

#### Key Terms

**heat capacity**: The 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 capacity**: The 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]\left(J\cdot K^{-1}\right)[/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\cdot\text{ 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\cdot 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 *[latex]75.2\frac{J}{\text{mol}\cdot 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 }\text{H}_2\text{O}}{\text{18 g}}=\text{2.0 mol H}_2\text{O}[/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}{\text{mol}\cdot K}\right)(10\;K)[/latex]

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

**Interactive: Seeing Specific Heat and Latent Heat**: Specific heat capacity is the measure of the heat energy required to raise the temperature of a given quantity of a substance by one kelvin. Latent heat of melting describes tœhe amount of heat required to melt a solid. When a solid is undergoing melting, the temperature basically remains constant until the entire solid is molten. The above simulation demonstrates the specific heat and the latent heat.

## Constant-Volume Calorimetry

Constant-volume calorimeters, such as bomb calorimeters, are used to measure the heat of combustion of a reaction.

### Learning Objectives

Describe how a bomb calorimeter works

### Key Takeaways

#### Key Points

- A bomb calorimeter is used to measure the change in internal energy, [latex]\Delta U[/latex], of a reaction. At constant volume, this is equal to
*q*, the heat of reaction._{V} - The calorimeter has its own heat capacity, which must be accounted for when doing calculations.

#### Key Terms

**bomb calorimeter**: A bomb calorimeter is a type of constant-volume calorimeter used in measuring the heat of combustion of a particular reaction.**calorie**: The amount of energy needed to raise the temperature of 1 gram of water by 1 °C. It is a non-SI unit of energy equivalent to approximately 4.18 Joules. A Calorie (with a capital C) = 1000 calories.

### The Bomb Calorimeter

Bomb calorimetry is used to measure the heat that a reaction absorbs or releases, and is practically used to measure the calorie content of food. A bomb calorimeter is a type of constant-volume calorimeter used to measure a particular reaction’s heat of combustion. For instance, if we were interested in determining the heat content of a sushi roll, for example, we would be looking to find out the number of calories it contains. In order to do this, we would place the sushi roll in a container referred to as the “bomb”, seal it, and then immerse it in the water inside the calorimeter. Then, we would evacuate all the air out of the bomb before pumping in pure oxygen gas (O_{2}). After the oxygen is added, a fuse would ignite the sample causing it to combust, thereby yielding carbon dioxide, gaseous water, and heat. As such, bomb calorimeters are built to withstand the large pressures produced from the gaseous products in these combustion reactions.

Once the sample is completely combusted, the heat released in the reaction transfers to the water and the calorimeter. The temperature change of the water is measured with a thermometer. The total heat given off in the reaction will be equal to the heat gained by the water and the calorimeter:

[latex]q_{rxn}=-q_{cal}[/latex]

Keep in mind that the heat gained by the calorimeter is the sum of the heat gained by the water, as well as the calorimeter itself. This can be expressed as follows:

[latex]q_{cal}=m_{\text{water}}C_{\text{water}}\Delta T+C_{cal}\Delta T[/latex]

where C_{water} denotes the *specific *heat capacity of the water [latex]\left(1 \frac{\text{cal}}{\text{g} ^{\circ}\text{C}}\right)[/latex], and C_{cal} is the heat capacity of the calorimeter (typically in [latex]\frac{\text{cal}}{^{\circ}\text{C}}[/latex]). Therefore, when running bomb calorimetry experiments, it is necessary to calibrate the calorimeter in order to determine *C _{cal}*.

Since the volume is constant for a bomb calorimeter, there is no pressure-volume work. As a result:

ΔU=q_{V}

where ΔU is the change in internal energy, and q_{V} denotes the heat absorbed or released by the reaction measured under conditions of *constant volume*. (This expression was previously derived in the “Internal Energy and Enthalpy ” section.) Thus, the total heat given off by the reaction is related to the change in internal energy (ΔU), not the change in enthalpy (ΔH) which is measured under conditions of *constant pressure*.

The value produced by such experiments does not completely reflect how our body burns food. For example, we cannot digest fiber, so obtained values have to be corrected to account for such differences between experimental (total) and actual (what the human body can absorb) values.

## Constant-Pressure Calorimetry

A constant-pressure calorimeter measures the change in enthalpy of a reaction at constant pressure.

### Learning Objectives

Discuss how a constant-pressure calorimeter works

### Key Takeaways

#### Key Points

- A constant- pressure calorimeter measures the change in enthalpy ( [latex]\Delta H[/latex] ) of a reaction occurring in solution, during which the pressure remains constant. Under these conditions, the change in enthalpy of the reaction is equal to the measured heat.
- Change in enthalpy can be calculated based on the change in temperature of the solution, its specific heat capacity, and mass.

#### Key Terms

**constant-pressure calorimeter**: Measures the change in enthalpy of a reaction occurring in solution, during which the pressure remains constant.**adiabatic**: Not allowing any transfer of heat energy; perfectly insulating.**coffee-cup calorimeter**: An example of constant-pressure calorimeter.

### Constant-Pressure Calorimetry

A constant-pressure calorimeter measures the change in enthalpy of a reaction occurring in a liquid solution. In that case, the gaseous pressure above the solution remains constant, and we say that the reaction is occurring under conditions of constant pressure. The heat transferred to/from the solution in order for the reaction to occur is equal to the change in enthalpy ([latex]\Delta H = q_P[/latex]), and a constant-pressure calorimeter thus measures this heat of reaction. In contrast, a bomb calorimeter ‘s volume is constant, so there is no pressure-volume work and the heat measured relates to the change in internal energy ([latex]\Delta U=q_V[/latex]).

A simple example of a constant-pressure calorimeter is a coffee-cup calorimeter, which is constructed from two nested Styrofoam cups and a lid with two holes, which allows for the insertion of a thermometer and a stirring rod. The inner cup holds a known amount of a liquid, usually water, that absorbs the heat from the reaction. The outer cup is assumed to be perfectly adiabatic, meaning that it does not absorb any heat whatsoever. As such, the outer cup is assumed to be a perfect insulator.

### Calculating Specific Heat

Data collected during a constant-pressure calorimetry experiment can be used to calculate the heat capacity of an unknown substance. We already know our equation relating heat (q), specific heat capacity (C), and the change in observed temperature ([latex]\Delta T[/latex]):

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

We will now illustrate how to use this equation to calculate the specific heat capacity of a substance.

### Examples

### Example 1

A student heats a 5.0 g sample of an unknown metal to a temperature of 207 [latex]^\circ[/latex]C, and then drops the sample into a coffee-cup calorimeter containing 36.0 g of water at 25.0 [latex]^\circ[/latex]C. After thermal equilibrium has been established, the final temperature of the water in the calorimeter is 26.0[latex]^\circ[/latex]C. What is the specific heat of the unknown metal? (The specific heat of water is 4.18 [latex]\frac {J} {g^\circ C}[/latex])

The walls of the coffee-cup calorimeter are assumed to be perfectly adiabatic, so we can assume that all of the heat from the metal was transferred to the water:

[latex]-q_{\text{metal}}=q_{\text{water}}[/latex]

Substituting in our above equation, we get:

[latex]-m_{\text{metal}}C_{\text{metal}} \Delta T_{\text{metal}}=m_{\text{water}}C_{\text{water}}\Delta T_{\text{water}}[/latex]

Then we can plug in our known values:

[latex]-(5.0\text{ g})C_{\text{metal}}(26.0^\circ\text{C}-207^\circ\text{C})=(36.0\text{ g})(4.18\; \frac {J}{\text{g}^\circ\text{C}})(26.0^\circ\text{C}-25.0^\circ\text{C})[/latex]

Solving for [latex]C_{\text{metal}}[/latex], we obtain

[latex]C_{metal}=0.166\; \frac {J} {g^\circ\text{C}}[/latex]

The specific heat capacity of the unknown metal is 0.166 [latex]\frac {J} {g ^\circ\text{C}}[/latex].

### Example 2

To determine the standard enthalpy of the reaction H^{+}(aq) + OH^{–}(aq) → H_{2}O(l), equal volumes of 0.1 M solutions of HCl and of NaOH can be combined initially at 25°C.

This process is exothermic and as a result, a certain amount of heat q_{P} will be released into the solution. The number of joules of heat released into each gram of the solution is calculated from the product of the rise in temperature and the specific heat capacity of water (assuming that the solution is dilute enough so that its specific heat capacity is *the same *as that of pure water’s). The total quantity of transferred heat can then be calculated by multiplying the result with the mass of the solution.

[latex]\Delta H=q_P = m_{\text{sol'n}}C_{\text{water}} \Delta T_{\text{sol'n}}[/latex]

Note that ΔH = q_{P} because the process is carried out at constant pressure.