# Heat Capacity

The *heat capacity* of a defined system is the amount of heat (usually expressed in calories, kilocalories, or joules) needed to raise the system's temperature by one degree (usually expressed in Celsius or Kelvin). It is expressed in units of thermal energy per degree temperature. To aid in the analysis of systems having certain specific dimensions, *molar heat capacity* and *specific heat capacity* can be used. To measure the heat capacity of a reaction, a calorimeter must be used. Bomb calorimeters are used for constant volume heat capacities, although a coffee-cup calorimeter is sufficient for a constant pressure heat capacity.

Definition: Molar Heat Capacity

The amount of heat needed to increase the temperature of one mole of a substance by one degree is the molar heat capacity. It is expressed in joules per moles per degrees Celsius (or Kelvin), \( \frac{Joules}{Moles\;^{\circ}C} \). For example, the molar heat capacity of lead is 26.65 \( \frac{Joules}{Moles\; C}\), which means that it takes 26.65 Joules of heat to raise 1 mole of lead by °C.

Definition: Specific Heat Capacity

The amount of heat needed to increase the temperature of one gram of a substance by one degree is the specific heat capacity. It is expressed in joules per gram per degree Celsius, \( \frac{Joules}{Grams \; ^{\circ}C} \). Because the specific heat of lead is 0.128 \( \frac{Joules}{Grams\;^{\circ}C}\), it 0.128 Joules of heat is required to raise one gram of lead by one °C.

### Quantity of Heat

The quantity of heat is a measurement of the amount of heat is present. The formula of quantity of heat, *q*, is equal to the mass of substance, *m*, multiplied with the specific heat and the change in temperature, \(\Delta {T}\). When the mass of substance is multiplied with the specific heat the product is equal to heat capacity, which is donated as \(C\).

\[q= \Delta{T} \times C \times m\]

Heat capacity, C, can never be negative for a mass or a substance and the specific heat of a substance can never be negative. Thus, if the change in temperature is negative, the initial temperature is more than the final temperature, then quantity of heat must be negative, for a negative number multiplied by a positive number equals a negative number. When the quantity of heat heat is negative heat the system is depleted of its heat; however, if the quantity of heat is positive then the system gains heat.

The total heat in a **closed system** must remain constant, which is represented by the equation

\[ q_{system}+ q_{surroundings}=0\]

This means that it is possible to set the quantity of heat of the system equal to the quantity of heat of the surroundings multiplied by negative one, which is used in the first calculation question of lab.

### Specific Latent Heat

There are two types of specific latent heat: vaporization and fusion. The specific latent heat of vaporization is defined as the quantity of heat energy that is necessary to raise one unit of weight (pounds or grams) with no change of temperature in the surroundings. Like the name implies, this specific latent heat quantifies the transfer of energy when a substance's state changes from liquid to gas or from gas to liquid. On the other hand, the specific heat of fusion is the quantity of heat that is necessary to raise one unit of weight without any change in temperature. This specific latent heat quantifies the transfer of energy when a substance's state changes from a solid to a liquid or from a liquid to a solid. Two formulas has be derived from this property:

\[q=m \times L\]

and* thus*

\[L=\dfrac{q}{m} \]

with

- \(q\)is the amount of heat increase or decrease as the state changes,
- \(m\) is the mass of the substance present, and
- \(L\) is the specific latent heat for that substance.

Example \(\PageIndex{1}\)

The specific heat capacity of water is 4.18 joules per gram per degree Celsius. How many joules of heat must be added to one gram of water to increase its temperature by 10 degrees Celsius?

**SOLUTION**

amount of heat=(mass of substance) x (specific heat capacity)x (change of temperature)

amount of heat=(1gram)(4.18 joules)/(grams °C)(10°C)

amount of heat=41.8 joules

Example \(\PageIndex{2}\)

In a calorimeter there is only water at room temperature (25°C). About 1.6 grams of ice are added to the system, and the temperature decreases to 1.2°C. The specific heat of water is 4.186 J/(g °C). What is the quantity of heat of calorimeter and the reaction?

**SOLUTION**

Because the change in temperature is given, the heat capacity of the calorimeter is the only unknown constant needed to solve for the quantity of heat of the calorimeter. However, the specific heat, *C*, of the calorimeter is equal to the specific heat of water, which is 4.186 J/(g °C). Now it's possible to solve for the quantity of heat.

1.

qcalorimeter= \(\Delta T_{}\) *C*

qcalorimeter= (1.2°C - 25°C) 4.186 J/(g °C)= -99.627 J

2. The formula for the quantity of heat of the calorimeter is:

qcalorimeter= -qreaction

-99.627 J= -qreaction

qreaction= 99.627 J

Example \(\PageIndex{3}\)

There is a coffee-cup calorimeter is filled with water at room temperature (25°C) as well as 2 grams of copper. If the two grams of copper are heated up to 46.3°C, what is the quantity of heat of the calorimeter and the copper?

**SOLUTION**

1. qcopper= \(\Delta T_{}\) *C*

qcopper= (46.3°C - 25°C) 4.186 J/(g °C)= 89.162 J

2. The formula for the quantity of heat of the calorimeter is:

qcopper= -qcalorimeter

89.162 J= -qcalorimeter

qcalorimeter= -99.627 J

Example \(\PageIndex{4}\)

If the quantity of heat of aluminum is 0.903 J/(g*°C) and the mass of the aluminum is 105 grams, what is the specific latent heat of the aluminum as it's state goes from solid to liquid?

**SOLUTION**

*L=Q/m*

*L= *(0.903 J/(g °C)) (105 g)*= *94.815 J/g= 94.815 kJ/kg.

The temperature (°C) is neglected because there is no change in temperature.

Example \(\PageIndex{5}\)

The specific latent heat of sulfur is 4.84 kJ/kg and the quantity of heat is 0.706 kJ/(kg*°C)

*Q=m L*

*m=L/Q*

*m=* (4.84 kJ/kg)/(0.706 kJ/(kg*°C)= 6.856 kg

The temperature (°C) is neglected because there is no change in temperature.

### References

- Petrucci, et al. General Chemistry: Principles & Modern Applications: AIE (Hardcover). Upper Saddle River: Pearson/Prentice Hall, 2007.
- Kotz, John C., Treichel, Paul M., and Townsend, John.
__Chemistry & Chemical Reactivity__. 7th Ed. Belmont: Thomson Higher Education, 2006. Print.

### Contributors

- Cameron Tracy, Ravneet Singh (UCD)