Heat Capacity: Study with Video Lessons, Practice Problems & Examples

Heat capacity measures the heat required to change a substance's temperature, with specific heat capacity (c) defined as the heat needed to raise 1 gram by 1 degree Celsius or Kelvin, and molar heat capacity (C) for 1 mole. The equations are q=mcΔT for specific heat and C=q⁄nΔT for molar heat. Understanding these concepts is crucial for applications in thermodynamics and chemical reactions.

Heat Capacity

So when we talk about heat capacity, we're talking about the application of heat to a substance. We're going to say that as we heat an object, its temperature increases because heat is directly proportional to its temperature change. The more heat I apply to something, the greater the temperature change will be. So here, to illustrate that, we say that q ∝ Δt. Just remember, that is the relationship between heat and temperature.

Heat Capacity Example 1

Here it says if the temperature of a water bath goes from 25 Kelvin to 50 Kelvin, what can be said about the amount of heat? So remember we said that heat, which is q, is directly proportional to change in temperature. Here our temperature is going from 25 Kelvin to 50 Kelvin, so it is being doubled in terms of Kelvin. And since they're directly proportional, what happens to one happens to the other. With our temperature doubling, that would mean that my heat would also have to double. This means that option a would be our correct answer.

Heat Capacity

Now, heat capacity, which uses capital C, is the amount of heat required to change the temperature of a weighted substance. The more heat that's applied to a substance, the greater its temperature change. It can also be analyzed in terms of specific heat capacity and molar heat capacity. With specific heat capacity, we use lowercase c, and it is the amount of heat required to change the temperature of 1 gram of a substance by 1 degree. That degree can be either in Kelvin or degrees Celsius. Here, molar heat capacity is just like heat capacity in terms that it uses a capital C. But with molar heat capacity, it's the amount of heat required to change the temperature of 1 mole of a substance by 1 degree, either in Kelvin or degrees Celsius. Think of molar heat capacity as being more focused in terms of the way we look at heat capacity, specifically concerning 1 mole of a substance.

Now, we're going to say here that when it comes to molar heat capacity, which is capital C, it is defined by the equation: C = q n ⋅ Δt Here, capital C equals our molar heat capacity in joules per moles times degrees Celsius or Kelvin. Q represents heat, T equals temperature in degrees Celsius, but the units of the molar heat capacity for temperature should match. So, if this is in Kelvin then the temperature should also be in Kelvin. And then, n is equal to our moles.

With our specific heat capacity, which uses lowercase c, it is defined by the equation: c = q m ⋅ Δt Here, lowercase c is our specific heat capacity in joules per grams times degrees Celsius or Kelvin. Again, q is heat. Temperature can also be in Celsius or Kelvin. To determine which one to use, you look at the units for your specific heat capacity and make sure they match. Lowercase m here just represents grams of our substance. So, just remember the difference between molar heat capacity and specific heat capacity.

Heat Capacity Example 2

Here the example says, if 15.7 grams of silver raises its temperature by 17.2 degrees Celsius when it absorbs 6,845.5 joules, what is its molar heat capacity? So, molar heat capacity uses capital C. It's equal to heat, which is \( q \) divided by moles \( n \) times change in temperature \(\Delta T\). In the question, it says that we're absorbing this much energy. That means that that's a positive \( q \). So that's positive 6,845.5 joules.

Next, we need moles, and we already have the change in temperature. They said that the temperature was risen by 17.2 degrees Celsius. So that's already our change in temperature. We need moles. We have here 15.7 grams of silver, which is \( g \). We have to change that to moles, so one mole of silver weighs 107.87 grams according to the periodic table. So that comes out to be 0.145548 moles of silver. Take those moles and plug it in. So when we do that, that's going to give me my molar heat capacity as 2,734.45 joules over moles times degrees Celsius. If we look at our values, this has 3 significant figures and this has 3 significant figures, so I could change this to \( 2.73 \times 10^3 \) joules over moles times degrees Celsius. So that would be the molar heat capacity for silver under these conditions.

Heat Capacity

Now by rearranging the specific heat capacity given above, we can solve for the amount of heat released or absorbed. Here our new specific heat capacity formula becomes q=m•c•Δt. For all of you pre-med track students, we usually say that this is equal to q=mcat, and we know that the MCAT is an important exam before you matriculate into medical school. So use that to help you remember it. So q=mcat is our new formula to help us determine and relate the specific heat capacity to the amount of heat absorbed or released in a chemical reaction.

Heat Capacity Example 3

Here it says, how much heat in kilojoules is released when 120 grams of water goes from 90 degrees Celsius to 45 degrees Celsius. The specific heat capacity of water is given as c=4.184Jg·C. So, we need to determine the heat which is q, they're giving us here the mass of the water which is m, we have our change in temperature here, and we have our specific heat capacity which is lowercase c. So q=mcΔT, plug in the grams of water given to us, plug in the specific heat capacity that's given to us, and remember the units for the specific heat capacity dictate the units for temperature. Since this is in Celsius it's good to have our temperatures in Celsius as well. ΔT=T_final-T_initial. So when we do all that, Celsius cancel out, grams cancel out, and we'll have joules of heat involved. When we plug that in, we're gonna get here negative 22593.6 joules. But here we want the answer in kilojoules, so 1kJ=103J. So here joules cancel out and we'll have kilojoules at the end. So that comes out to negative 22.5936 kilojoules. Here, this has 4 significant figures, 1 significant figure, 2 significant figures. Specific heat capacity we could use it since it's given to us as a value, but we don't have to. Here we could just go with 1 significant figure, but I feel like that's too much rounding, so let's just go with 2 significant figures based on 45. So it's gonna be negative 23, roughly, kilojoules of heat are released. So here using q=mcΔT we're able to isolate the heat that's involved in the releasing by the water molecule going from 90 degrees Celsius to 45 degrees Celsius.

A sample of copper absorbs 3.53 kJ of heat, which increases the temperature by 25 ºC, determine the mass (in kg) of the copper sample if the specific heat capacity of copper is 0.385 J / g  ºC.

• A. 0.73 kg
• B. 0.35 kg
• C. 0.37 kg
• D. 0.53 kg

Based on their given specific heat capacities which compound would show the greatest temperature change upon absorbing 25.0 J of heat? 

• A. 250.0 g Al
• B. 250.0 g Cu
• C. 250.0 g ethanol
• D. 250.0 g wood

50.00 g of heated metal ore is placed into an insulated beaker containing 822.5 g of water. Once the metal heats up the final temperature of the water is 32.08 ºC. If the metal gains 14.55 kJ of energy, what is the initial temperature of the water? The specific heat capacity of copper is 4.184 J / g  ºC.

• A. 23.86 °C
• B. 32.86 °C
• C. 63.08 °C
• D. 36.31 °C

Here’s what students ask on this topic:

What is the difference between specific heat capacity and molar heat capacity?

Specific heat capacity (c) is the amount of heat required to raise the temperature of 1 gram of a substance by 1 degree Celsius or Kelvin. It is expressed in units of J/g·°C or J/g·K. The formula is $q=mc\mathrm{\Delta T}$, where $q$ is heat, $m$ is mass, and $\mathrm{\Delta T}$ is the temperature change. Molar heat capacity (C) is the amount of heat required to raise the temperature of 1 mole of a substance by 1 degree Celsius or Kelvin. It is expressed in units of J/mol·°C or J/mol·K. The formula is $C=\frac{q}{n}\mathrm{\Delta T}$, where $q$ is heat, $n$ is moles, and $\mathrm{\Delta T}$ is the temperature change.

How do you calculate the amount of heat absorbed or released using specific heat capacity?

To calculate the amount of heat absorbed or released using specific heat capacity, you can use the formula $q=mc\mathrm{\Delta T}$. Here, $q$ represents the heat absorbed or released, $m$ is the mass of the substance in grams, $c$ is the specific heat capacity in J/g·°C or J/g·K, and $\mathrm{\Delta T}$ is the change in temperature in degrees Celsius or Kelvin. By multiplying these values, you can determine the total heat energy involved in the process.

Why is it important to match the units of temperature when using heat capacity formulas?

It is crucial to match the units of temperature when using heat capacity formulas to ensure consistency and accuracy in calculations. For specific heat capacity, the formula is $q=mc\mathrm{\Delta T}$, and for molar heat capacity, it is $C=\frac{q}{n}\mathrm{\Delta T}$. If the temperature change $\mathrm{\Delta T}$ is in degrees Celsius, the heat capacity should also be in units that correspond to degrees Celsius (e.g., J/g·°C or J/mol·°C). Similarly, if the temperature change is in Kelvin, the heat capacity should be in units that correspond to Kelvin (e.g., J/g·K or J/mol·K). This ensures that the calculated heat energy is accurate and meaningful.

What is the formula for molar heat capacity and how is it used?

The formula for molar heat capacity (C) is $C=\frac{q}{n}\mathrm{\Delta T}$, where $q$ is the amount of heat absorbed or released, $n$ is the number of moles of the substance, and $\mathrm{\Delta T}$ is the change in temperature. This formula is used to determine the heat capacity per mole of a substance, which is useful in understanding how much heat is required to change the temperature of a given amount of substance. It is particularly important in chemical reactions and thermodynamic calculations.

How does the concept of heat capacity apply to chemical reactions?

Heat capacity is crucial in chemical reactions because it helps determine the amount of heat absorbed or released during the reaction. By knowing the specific or molar heat capacity of the reactants and products, you can calculate the heat change using the formulas $q=mc\mathrm{\Delta T}$ for specific heat capacity and $C=\frac{q}{n}\mathrm{\Delta T}$ for molar heat capacity. This information is essential for understanding reaction energetics, predicting temperature changes, and designing processes that require precise thermal management. It also aids in calculating enthalpy changes and understanding the thermodynamic properties of the substances involved.