# Heat and temperature

What heat means in thermodynamics, and how we can calculate heat using the heat capacity.

## Key points

• Heat, $\text q$, is thermal energy transferred from a hotter system to a cooler system that are in contact.
• Temperature is a measure of the average kinetic energy of the atoms or molecules in the system.
• The zeroth law of thermodynamics says that no heat is transferred between two objects in thermal equilibrium; therefore, they are the same temperature.
• We can calculate the heat released or absorbed using the specific heat capacity $\text C$, the mass of the substance $\text m$, and the change in temperature $\Delta \text T$ in the equation:
$\text q = \text {m} \times \text C \times \Delta \text T$

## Heat in thermodynamics

What contains more heat, a cup of coffee or a glass of iced tea? In chemistry class, that would be a trick question (sorry!). In thermodynamics, heat has a very specific meaning that is different from how we might use the word in everyday speech. Scientists define heat as thermal energy transferred between two systems at different temperatures that come in contact. Heat is written with the symbol q or Q, and it has units of Joules ($\text J$).
Three melting ice cubes in a puddle of water on a mirrored surface.
Heat is transferred from the surroundings to the ice, causing the phase change from ice to water. Photo of ice cubes from flickr, CC BY 2.0.
Heat is sometimes called a process quantity, because it is defined in the context of a process by which energy can be transferred. We don't talk about a cup of coffee containing heat, but we can talk about the heat transferred from the cup of hot coffee to your hand. Heat is also an extensive property, so the change in temperature resulting from heat transferred to a system depends on how many molecules are in the system.

## Relationship between heat and temperature

Heat and temperature are two different but closely related concepts. Note that they have different units: temperature typically has units of degrees Celsius ($^\circ\text C$) or Kelvin ($\text K$), and heat has units of energy, Joules ($\text J$). Temperature is a measure of the average kinetic energy of the atoms or molecules in the system. The water molecules in a cup of hot coffee have a higher average kinetic energy than the water molecules in a cup of iced tea, which also means they are moving at a higher velocity. Temperature is also an intensive property, which means that the temperature doesn't change no matter how much of a substance you have (as long as it is all at the same temperature!). This is why chemists can use the melting point to help identify a pure substance$-$the temperature at which it melts is a property of the substance with no dependence on the mass of a sample.
On an atomic level, the molecules in each object are constantly in motion and colliding with each other. Every time molecules collide, kinetic energy can be transferred. When the two systems are in contact, heat will be transferred through molecular collisions from the hotter system to the cooler system. The thermal energy will flow in that direction until the two objects are at the same temperature. When the two systems in contact are at the same temperature, we say they are in thermal equilibrium.

## Zeroth law of thermodynamics: Defining thermal equilibrium

The zeroth law of thermodynamics defines thermal equilibrium within an isolated system. The zeroth law says when two objects at thermal equilibrium are in contact, there is no net heat transfer between the objects; therefore, they are the same temperature. Another way to state the zeroth law is to say that if two objects are both separately in thermal equilibrium with a third object, then they are in thermal equilibrium with each other.
The zeroth law allows us to measure the temperature of objects. Any time we use a thermometer, we are using the zeroth law of thermodynamics. Let's say we are measuring the temperature of a water bath. In order to make sure the reading is accurate, we usually want to wait for the temperature reading to stay constant. We are waiting for the thermometer and the water to reach thermal equilibrium! At thermal equilibrium, the temperature of the thermometer bulb and the water bath will be the same, and there should be no net heat transfer from one object to the other (assuming no other loss of heat to the surroundings).

## Heat capacity: Converting between heat and change in temperature

How can we measure heat? Here are some things we know about heat so far:
• When a system absorbs or loses heat, the average kinetic energy of the molecules will change. Thus, heat transfer results in a change in the system's temperature as long as the system is not undergoing a phase change.
• The change in temperature resulting from heat transferred to or from a system depends on how many molecules are in the system.
We can use a thermometer to measure the change in a system's temperature. How can we use the change in temperature to calculate the heat transferred?
In order to figure out how the heat transferred to a system will change the temperature of the system, we need to know at least $2$ things:
• The number of molecules in the system
• The heat capacity of the system
The heat capacity tells us how much energy is needed to change the temperature of a given substance assuming that no phase changes are occurring. There are two main ways that heat capacity is reported. The specific heat capacity (also called specific heat), represented by the symbol $\text c$ or $\text C$, is how much energy is needed to increase the temperature of one gram of a substance by $1~^{\circ}\text C$ or $1\,\text K$. Specific heat capacity usually has units of $\dfrac{\text J}{\text{grams}\cdot\text K}$. The molar heat capacity, $\text C_\text m$ or $\text C_{\text{mol}}$, measures the amount of thermal energy it takes to raise the temperature of one mole of a substance by $1~^{\circ}\text C$ or $1\,\text K$, and it usually has units of $\dfrac{\text J}{\text{mol}\cdot\text K}$. For example, the heat capacity of lead might be given as the specific heat capacity, $0.129\,\dfrac{\text J}{\text{g}\cdot\text K}$, or the molar heat capacity, $26.65\,\dfrac{\text J}{\text{mol}\cdot\text K}$.
Let's think about what is happening on a molecular level when we add thermal energy to some molecules. The thermal energy can be stored as vibrations and rotations between atoms within a molecule, which does not significantly increase the temperature of the system. The energy can also be used to disrupt intermolecular interactions and increase the velocity of the entire molecule, which increases the translational kinetic energy of the molecule.
Temperature is primarily a measure of the translational kinetic energy of the system. Depending on the molecular structure and intermolecular interactions, different substances can store different amounts of thermal energy as vibrations and rotations before the temperature increases.

## Calculating $\text q$ using the heat capacity

We can use the heat capacity to determine the heat released or absorbed by a material using the following formula:
$\text q = \text {m} \times \text C \times \Delta \text T$
where $\text{m}$ is the mass of the substance (in grams), $\text{C}$ is the specific heat capacity, and $\Delta \text T$ is the change in temperature during the heat transfer. Note that both mass and specific heat capacity can only have positive values, so the sign of $\text q$ will depend on the sign of $\Delta \text T$. We can calculate $\Delta \text T$ using the following equation:
$\Delta \text T=\text T_{\text{final}}-\text T_{\text{initial}}$
where $\text T_{\text{final}}$ and $\text T_{\text{initial}}$ can have units of either $~^{\circ}\text C$ or $\text K$. Based on this equation, if $\text q$ is positive (energy of the system increases), then our system increases in temperature and $\text T_{\text{final}}>\text T_{\text{initial}}$. If $\text q$ is negative (energy of the system decreases), then our system's temperature decreases and $\text T_{\text{final}}<\text T_{\text{initial}}$.

## Example problem: Cooling a cup of tea

Let's say that we have $250\,\text{mL}$ of hot tea which we would like to cool down before we try to drink it. The tea is currently at $370\,\text K$, and we'd like to cool it down to $350\,\text K$. How much thermal energy has to be transferred from the tea to the surroundings to cool the tea?
A cup of black tea with a slice of lemon in a white teacup with a saucer.
The hot tea will transfer heat to the surroundings as it cools. Photo from Photozou, CC BY-NC-ND 2.5
We are going to assume that the tea is mostly water, so we can use the density and heat capacity of water in our calculations. The specific heat capacity of water is $4.18\,\dfrac{\text J}{\text g \cdot \text K}$, and the density of water is $1.00\,\dfrac{\text g}{\text {mL}}$. We can calculate the energy transferred in the process of cooling the tea using the following steps:

### 1. Calculate the mass of the substance

We can calculate the mass of the tea/water using the volume and density of water:
$\text m=250\,\cancel{\text {mL}} \times 1.00\,\dfrac{\text g}{\cancel{\text {mL}}}=250\,\text g$

### 2. Calculate the change in temperature, $\Delta \text T$

We can calculate the change in temperature, $\Delta \text T$, from the initial and final temperatures:
\begin{aligned}\Delta \text T&=\text T_{\text{final}}-\text T_{\text{initial}}\\ \\ &=350\,\text K-370\,\text K\\ \\ &=-20\,\text K\end{aligned}
Since the temperature of the tea is decreasing and $\Delta \text T$ is negative, we would expect $\text q$ to also be negative since our system is losing thermal energy.

### 3. Solve for $\text q$

Now we can solve for the heat transferred from the hot tea using the equation for heat:
\begin{aligned}\text q &= \text {m} \times \text C \times \Delta \text T\\ &=250\,\cancel{\text g} \times4.18\,\dfrac{\text J}{\cancel{\text g} \cdot \cancel{\text K}} \times -20\,\cancel{\text K}\\ &=-21000\,\text J\end{aligned}
Thus, we calculated that the tea will transfer $21000\,\text J$ of energy to the surroundings when it cools down from $370\,\text K$ to $350\,\text K$.

## Conclusions

In thermodynamics, heat and temperature are closely related concepts with precise definitions.
• Heat, $\text q$, is thermal energy transferred from a hotter system to a cooler system that are in contact.
• Temperature is a measure of the average kinetic energy of the atoms or molecules in the system.
• The zeroth law of thermodynamics says that no heat is transferred between two objects in thermal equilibrium; therefore, they are the same temperature.
• We can calculate the heat released or absorbed using the specific heat capacity $\text C$, the mass of the substance $\text m$, and the change in temperature $\Delta \text T$ in the following equation:
$\text q = \text {m} \times \text C \times \Delta \text T$

1. Heat and Work” from Boundless Chemistry, CC BY 4.0.
2. "The Zeroth Law of Thermodynamics" from Boundless Chemistry, CC BY 4.0.
3. "Heat Capacity" from UC Davis ChemWIki, CC BY-NC-SA 3.0 US.