Molar specific heat capacities
Molar Specific Heat Capacities
Molar specific heat capacity is an important concept in thermodynamics that describes the amount of heat required to raise the temperature of one mole of a substance by one degree Celsius (or one Kelvin). It is a property that is intrinsic to the material and can vary depending on the conditions, such as constant volume or constant pressure.
Definitions
Specific Heat Capacity at Constant Volume ($C_V$)
This is the amount of heat required to raise the temperature of one mole of a substance by one degree Celsius at constant volume. It is expressed in units of joules per mole per Kelvin (J/mol·K).
Specific Heat Capacity at Constant Pressure ($C_P$)
This is the amount of heat required to raise the temperature of one mole of a substance by one degree Celsius at constant pressure. It is also expressed in units of joules per mole per Kelvin (J/mol·K).
Formulas
The heat added or removed from a substance can be calculated using the formula:
$$ q = nC\Delta T $$
where:
- $q$ is the heat added or removed (in joules, J),
- $n$ is the number of moles,
- $C$ is the molar specific heat capacity (at constant volume $C_V$ or constant pressure $C_P$),
- $\Delta T$ is the change in temperature (in Kelvin, K or degrees Celsius, °C).
The relationship between the molar specific heat capacities at constant pressure and constant volume for an ideal gas is given by the equation:
$$ C_P - C_V = R $$
where:
- $R$ is the ideal gas constant, which is approximately 8.314 J/mol·K.
Differences and Important Points
| Property | Specific Heat at Constant Volume ($C_V$) | Specific Heat at Constant Pressure ($C_P$) |
|---|---|---|
| Definition | Heat required to raise the temperature of one mole of a substance by 1°C at constant volume. | Heat required to raise the temperature of one mole of a substance by 1°C at constant pressure. |
| Formula | $q = nC_V\Delta T$ | $q = nC_P\Delta T$ |
| Units | J/mol·K | J/mol·K |
| For Ideal Gases | $C_V$ is typically lower than $C_P$ because no work is done against external pressure. | $C_P$ is higher than $C_V$ because the system does work on the surroundings by expanding. |
| Relationship | $C_P - C_V = R$ | $C_P = C_V + R$ |
Examples
Example 1: Calculating Heat Added
Suppose you have 2 moles of an ideal gas at constant pressure, and you want to increase its temperature from 25°C to 35°C. The molar specific heat capacity at constant pressure for the gas is $29 J/mol·K$. How much heat is required?
Using the formula:
$$ q = nC_P\Delta T $$
we get:
$$ q = 2 \text{ moles} \times 29 \frac{J}{mol·K} \times (35°C - 25°C) $$ $$ q = 2 \times 29 \times 10 $$ $$ q = 580 J $$
So, 580 joules of heat are required to raise the temperature of the gas by 10°C at constant pressure.
Example 2: Relationship Between $C_P$ and $C_V$
For an ideal gas, if the molar specific heat capacity at constant volume ($C_V$) is 20.8 J/mol·K, what is the molar specific heat capacity at constant pressure ($C_P$)?
Using the relationship:
$$ C_P = C_V + R $$
and substituting the values:
$$ C_P = 20.8 \frac{J}{mol·K} + 8.314 \frac{J}{mol·K} $$ $$ C_P = 29.114 \frac{J}{mol·K} $$
Therefore, the molar specific heat capacity at constant pressure is approximately 29.114 J/mol·K.
Conclusion
Understanding molar specific heat capacities is crucial for solving problems in thermodynamics related to heat transfer and temperature changes in substances. The distinction between $C_V$ and $C_P$ is particularly important for gases, where the ability to do work on the surroundings (or have work done on them) at constant pressure must be taken into account. Remembering the relationship between $C_P$ and $C_V$ and the ideal gas constant is a key part of working with these concepts in a practical context.