Physics and Mathematics
Molar Specific Heat or Molar Heat Capacity
1. Concept Overview
The molar specific heat or molar heat capacity of a substance is defined as the amount of heat required to raise the temperature of 1 mole of the substance by 1 Kelvin (or 1°C).
It is an extension of the concept of specific heat, but here we refer to one mole of the substance instead of one kilogram.
Thus, it represents how much heat is needed to change the temperature of Avogadro’s number of molecules of a substance by one degree.
2. Explanation and Mathematical Derivation
Let:
- [Q] = amount of heat supplied
- [n] = number of moles of the substance
- [C] = molar heat capacity
- [\Delta T] = rise in temperature
Then, the heat supplied is given by:
[Q = nC\Delta T]
Hence, the molar heat capacity is defined as:
[C = \dfrac{Q}{n\Delta T}]
Relation with Specific Heat (c)
If [M] is the molar mass (in kg/mol) and [c] is the specific heat of the substance,
then the relation between molar heat capacity and specific heat is:
[C = M c]
That is, molar heat capacity = specific heat × molar mass.
Types of Molar Heat Capacities (for gases)
For gases, the molar heat capacity depends on the thermodynamic process:
- At Constant Volume (Cᵥ):
[Q = nC_v \Delta T]
No work is done since volume is constant. - At Constant Pressure (Cₚ):
[Q = n C_p \Delta T]
Work is done by the gas as it expands.
The relationship between the two for an ideal gas is:
[C_p – C_v = R]
where [R = 8.314 J mol^{-1} K^{-1}] is the universal gas constant.
3. Dimensions and Units
| Quantity | Symbol | SI Unit | Dimensions |
|---|---|---|---|
| Heat Energy | [Q] | J | [M L² T⁻²] |
| Number of Moles | [n] | mol | — |
| Temperature Change | [\Delta T] | K | [K] |
| Molar Heat Capacity | [C] | J·mol⁻¹·K⁻¹ | [L² T⁻² K⁻¹ mol⁻¹] |
4. Key Features
- Molar heat capacity is the heat capacity per mole of a substance.
- It depends on the nature and phase of the substance (solid, liquid, gas).
- For ideal gases, [C_p] and [C_v] are constant.
- The difference [C_p – C_v = R] holds for all ideal gases.
- Solids and liquids generally have only one molar heat value (no distinction between constant pressure and volume).
- Units: [1 J mol^{-1} K^{-1}] = [1 N m mol^{-1} K^{-1}].
- Molar heat capacity gives direct information about microscopic degrees of freedom of molecules.
- It forms the foundation for the equipartition of energy theorem and specific heat laws in gases.
5. Important Formulas to Remember
| Formula | Description |
|---|---|
| [Q = n C \Delta T] | Basic definition |
| [C = \dfrac{Q}{n \Delta T}] | Molar heat capacity |
| [C = M c] | Relation with specific heat |
| [C_p – C_v = R] | Mayer’s relation for ideal gases |
| [Q = n C_p \Delta T] or [Q = n C_v \Delta T] | Heat supplied under constant pressure or volume |
6. Conceptual Questions with Solutions
1. What is molar specific heat?
It is the amount of heat required to raise the temperature of 1 mole of a substance by 1 K.
2. Write the SI unit of molar specific heat.
[J·mol⁻¹·K⁻¹].
3. How is molar heat related to specific heat?
[C = M c], where [M] = molar mass and [c] = specific heat.
4. What is the relation between Cₚ and Cᵥ for ideal gases?
[\; C_p – C_v = R \;]
5. Why does Cₚ exceed Cᵥ?
At constant pressure, the gas does external work while expanding, so more heat is needed.
6. What is the molar heat capacity of an ideal monoatomic gas at constant volume?
[\; C_v = \dfrac{3}{2} R \;]
7. What is its value at constant pressure?
[C_p] [= C_v + R] [= \dfrac{5}{2} R]
8. What does a high molar heat indicate?
It means the substance needs more energy to increase its temperature by 1 K per mole.
9. What are the units of R (gas constant)?
[R = 8.314 J mol^{-1} K^{-1}]
10. Can solids have different molar heat capacities at constant pressure and volume?
No, because solids expand very little; [C_p \approx C_v].
11. For a diatomic gas, what are the theoretical values of Cᵥ and Cₚ?
[C_v] [= \dfrac{5}{2} R \quad C_p] [= \dfrac{7}{2} R]
12. For a polyatomic gas?
[C_v] [= 3R \quad C_p = 4R]
13. What is the physical significance of Cᵥ?
It measures the energy required to increase the internal energy of the gas per mole per K when volume is constant.
14. Why is Cᵥ smaller than Cₚ?
Because at constant volume, no work is done, so less heat is needed.
15. What is the molar heat capacity of water?
Approximately [C = 75.4 J mol^{-1} K^{-1}].
7. FAQ / Common Misconceptions
1. Are Cₚ and Cᵥ the same for solids and liquids?
Yes, nearly the same because volume change is negligible.
2. Does molar heat depend on temperature?
Slightly — especially for gases, it can vary with temperature and molecular structure.
3. Can molar heat capacity be negative?
No, because increasing temperature always requires positive heat input.
4. Is molar heat capacity the same for all gases?
No, it depends on degrees of freedom of the gas molecules.
5. What does R represent in [C_p – C_v = R]?
The universal gas constant — same for all gases.
6. Does molar heat capacity depend on mass?
No, it depends only on moles, not total mass.
7. Why does hydrogen have a different molar heat than oxygen?
Because of different molecular structures and degrees of freedom.
8. Are Cₚ and Cᵥ constant for real gases?
Not exactly; they vary slightly with pressure and temperature.
9. Is Cᵥ always less than Cₚ?
Yes, always — since extra work is done at constant pressure.
10. Does [C_p – C_v = R] apply to solids?
No, it applies only to **ideal gases**.
8. Practice Questions with Step-by-Step Solutions
Q1. Calculate the molar heat capacity of copper given: [c = 385 J kg^{-1} K^{-1}], [M = 0.0635 kg mol^{-1}].
Solution:
[C = M c] [= 0.0635 \times 385] [= 24.4 J mol^{-1} K^{-1}]
Molar heat of copper = 24.4 J·mol⁻¹·K⁻¹
Q2. Find the heat required to raise the temperature of 2 mol of nitrogen by 50 K at constant pressure.
Given: [C_p = \dfrac{7}{2} R].
Solution:
[Q = n C_p \Delta T] [= 2 \times \dfrac{7}{2} \times 8.314 \times 50] [= 2 \times 3.5 \times 8.314 \times 50] [= 29100 J]
Heat required = 29.1 kJ
Q3. For 1 mol of helium gas, calculate [C_p] and [C_v].
Solution:
Helium is monoatomic.
[C_v] [= \dfrac{3}{2} R] [= 12.47 J mol^{-1} K^{-1}]
[C_p] [= C_v + R] [= \dfrac{5}{2} R] [= 20.79 J mol^{-1} K^{-1}]
Q4. If 500 J of heat increases the temperature of 0.5 mol of gas by 10 K at constant volume, find Cᵥ.
Solution:
[C_v] [= \dfrac{Q}{n \Delta T}] [= \dfrac{500}{0.5 \times 10}] [= 100 J mol^{-1} K^{-1}]
Q5. A diatomic gas has [C_p = 29.1 J mol^{-1} K^{-1}]. Find [C_v] and verify Mayer’s relation.
Solution:
[C_v] [= C_p – R] [= 29.1 – 8.314] [= 20.8 J mol^{-1} K^{-1}]
Verified that [C_p – C_v = R].
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