Heat Capacity: Videos & Practice Problems

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°C, 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 thermochemistry and calorimetry applications.

Heat Capacity

Heat capacity is a fundamental concept in thermodynamics that describes how a substance responds to the application of heat. When heat is added to an object, its temperature increases, demonstrating a direct relationship between the amount of heat applied and the resulting temperature change. This relationship can be expressed mathematically as:

$$ q \propto \Delta T $$

In this equation, q represents the heat added to the substance, and ΔT denotes the change in temperature. The proportionality indicates that as the heat (q) increases, the temperature change (ΔT) also increases, assuming the substance's properties remain constant. Understanding this relationship is crucial for calculating heat capacity, which is defined as the amount of heat required to change the temperature of a given mass of a substance by one degree Celsius (or Kelvin).

Heat Capacity Example 1

When analyzing the relationship between heat and temperature change, it's essential to understand that heat (denoted as q) is directly proportional to the change in temperature. In this scenario, the temperature of a water bath increases from 25 Kelvin to 50 Kelvin. This change can be calculated as follows:

Change in temperature, ΔT = Final Temperature - Initial Temperature = 50 K - 25 K = 25 K.

Since the temperature has effectively doubled (from 25 K to 50 K), we can infer that the amount of heat must also double, given their direct proportionality. Therefore, if the temperature increases, the heat absorbed or released will correspondingly increase. This leads us to conclude that the correct answer is that the amount of heat will also double as the temperature changes from 25 Kelvin to 50 Kelvin.

Heat Capacity

Heat capacity, denoted by the capital letter C, refers to the amount of heat required to change the temperature of a given mass of a substance. The relationship is straightforward: the more heat applied, the greater the temperature change observed. This concept can be further divided into specific heat capacity and molar heat capacity, each serving a distinct purpose in thermodynamics.

Specific heat capacity, represented by the lowercase c, quantifies the heat needed to raise the temperature of 1 gram of a substance by 1 degree Celsius (°C) or 1 Kelvin (K). This measurement is crucial for understanding how different materials respond to heat. In contrast, molar heat capacity, indicated by the capital C, measures the heat required to change the temperature of 1 mole of a substance by 1 degree Celsius or Kelvin. This distinction makes molar heat capacity more focused on the behavior of substances at the molecular level.

The formula for molar heat capacity is expressed as:

\( C = \frac{q}{n \cdot \Delta t} \)

In this equation, C is the molar heat capacity in joules per mole per degree Celsius (J/mol·°C) or joules per mole per Kelvin (J/mol·K), q represents the heat added, n is the number of moles, and Δt is the change in temperature. It is essential that the temperature units used in the calculation match those of the molar heat capacity.

On the other hand, the specific heat capacity is defined by the formula:

\( c = \frac{q}{m \cdot \Delta t} \)

Here, c is the specific heat capacity in joules per gram per degree Celsius (J/g·°C) or joules per gram per Kelvin (J/g·K), m is the mass of the substance in grams, and Δt again represents the temperature change. As with molar heat capacity, it is crucial to ensure that the temperature units correspond to those used in the specific heat capacity measurement.

Understanding the differences between molar heat capacity and specific heat capacity is vital for accurately calculating heat transfer in various chemical and physical processes.

Heat Capacity Example 2

To determine the molar heat capacity of silver, we start with the formula for molar heat capacity, represented as \( C \), which is defined as:

\[ C = \frac{q}{n \Delta T} \]

In this equation, \( q \) is the heat absorbed, \( n \) is the number of moles, and \( \Delta T \) is the change in temperature. In this scenario, silver absorbs \( 6,845.5 \) joules of energy, indicating that \( q = +6,845.5 \, \text{J} \). The temperature increase is given as \( 17.2 \, \text{°C} \), which serves as our \( \Delta T \).

Next, we need to convert the mass of silver from grams to moles. The mass of silver is \( 15.7 \, \text{g} \), and using the molar mass of silver, which is \( 107.87 \, \text{g/mol} \), we calculate the number of moles \( n \) as follows:

\[ n = \frac{15.7 \, \text{g}}{107.87 \, \text{g/mol}} \approx 0.145548 \, \text{mol} \]

Now, substituting the values into the molar heat capacity formula:

\[ C = \frac{6,845.5 \, \text{J}}{0.145548 \, \text{mol} \times 17.2 \, \text{°C}} \]

Calculating this gives:

\[ C \approx 2734.45 \, \text{J/(mol} \cdot \text{°C)} \]

Considering significant figures, we round the result to three significant figures, yielding:

\[ C \approx 2.73 \times 10^3 \, \text{J/(mol} \cdot \text{°C)} \]

This value represents the molar heat capacity of silver under the specified conditions.

Heat Capacity

In thermodynamics, the concept of specific heat capacity is crucial for understanding how substances absorb or release heat. The relationship can be expressed through the formula:

\( q = mc\Delta T \)

In this equation, \( q \) represents the amount of heat absorbed or released, \( m \) is the mass of the substance, \( c \) is the specific heat capacity, and \( \Delta T \) is the change in temperature. A helpful mnemonic for remembering this formula is to associate it with the MCAT, a significant exam for pre-med students, leading to the simplified expression:

\( q = mc\Delta T \) or \( q = m \cdot c \cdot \Delta T \)

This formula allows you to calculate the heat transfer in various chemical reactions, making it an essential tool in both chemistry and physics. Understanding how to manipulate this equation is vital for predicting how substances will behave under different thermal conditions.

Heat Capacity Example 3

To calculate the heat released when 120 grams of water cools from 90 degrees Celsius to 45 degrees Celsius, we can use the formula for heat transfer:

q = mcΔT

In this equation, q represents the heat absorbed or released (in joules), m is the mass of the substance (in grams), c is the specific heat capacity (in joules per gram per degree Celsius), and ΔT is the change in temperature (in degrees Celsius).

Given that:

• Mass of water, m = 120 g
• Specific heat capacity of water, c = 4.184 J/g°C
• Initial temperature, T_initial = 90°C
• Final temperature, T_final = 45°C

First, we calculate the change in temperature:

ΔT = T_final - T_initial = 45°C - 90°C = -45°C

Now, substituting the values into the heat transfer equation:

q = (120 g)(4.184 J/g°C)(-45°C)

Calculating this gives:

q = -22593.6 J

To convert joules to kilojoules, we use the conversion factor where 1 kilojoule = 1000 joules:

q = -22593.6 J × (1 kJ / 1000 J) = -22.5936 kJ

Considering significant figures, we round the result to two significant figures based on the precision of the given values:

q ≈ -23 kJ

This indicates that approximately 23 kilojoules of heat are released as the water cools 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

Answer: (C)

Based on their given specific heat capacities which substance 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

Answer: (B)

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

Answer: (D)

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°C or 1 K. It is expressed in units of J/g·°C or J/g·K. Molar heat capacity (C), on the other hand, is the amount of heat required to raise the temperature of 1 mole of a substance by 1°C or 1 K. It is expressed in units of J/mol·°C or J/mol·K. The key difference lies in the amount of substance being considered: specific heat capacity deals with mass (grams), while molar heat capacity deals with the number of moles.

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

To calculate the amount of heat (q) absorbed or released using specific heat capacity (c), you can use the formula:

$q=mc\mathrm{\Delta T}$

Here, $m$ is the mass of the substance in grams, $c$ is the specific heat capacity, and $\mathrm{\Delta T}$ is the change in temperature in °C or K. This formula helps determine the heat exchanged during a temperature change.

Why is heat capacity important in thermochemistry?

Heat capacity is crucial in thermochemistry because it helps quantify the amount of heat energy required to change the temperature of a substance. This information is essential for understanding and predicting the energy changes in chemical reactions and physical processes. By knowing the heat capacity, scientists can calculate the heat absorbed or released during reactions, which is vital for designing experiments, industrial processes, and understanding natural phenomena.

What units are used for specific heat capacity and molar heat capacity?

Specific heat capacity (c) is typically measured in units of J/g·°C or J/g·K, which represent the amount of heat required to raise the temperature of 1 gram of a substance by 1°C or 1 K. Molar heat capacity (C) is measured in units of J/mol·°C or J/mol·K, indicating the amount of heat needed to raise the temperature of 1 mole of a substance by 1°C or 1 K. The choice of units depends on whether the measurement is based on mass or moles.

How does the formula for molar heat capacity differ from that of specific heat capacity?

The formula for molar heat capacity (C) is:

$C=\frac{q}{n\mathrm{\Delta T}}$

Here, $q$ is the heat absorbed or released, $n$ is the number of moles, and $\mathrm{\Delta T}$ is the change in temperature. For specific heat capacity (c), the formula is:

$c=\frac{q}{m\mathrm{\Delta T}}$

Here, $m$ is the mass in grams. The main difference is that molar heat capacity uses moles, while specific heat capacity uses mass.