Key Differences Between Cp and Cv in Thermodynamics
The heat capacity of a substance, commonly abbreviated as thermal capacity (capital C), is a measure of the amount of heat needed to change the temperature of the substance by a specific amount. Heat capacity is measured in SI units and is referred to as the joules per kelvin (J/K) unit.
In a heat capacity calculation, the amount of heat energy transferred to an object (symbol C) is proportional to the amount of temperature increase that results.
\[C=Q\Delta T.C=Q\Delta T\].
A system's heat capacity scales with its size since it is an extensive property. When a sample contains twice as much substance as another, it takes twice as much heat (Q) to achieve the same temperature change (ΔT). It would take 2,000 J to heat a second iron block with twice the mass of a first iron block if it took 1,000 J to heat a block of iron.
The Measurement of Heat Capacity
It is not always possible to predict the capacity of a system in terms of heat. This depends more on the state variables of the thermodynamic system under discussion. The amount of change in volume or pressure is dependent upon many factors, including the temperature itself, the pressures that are in the system, and how those pressures have changed while the system has been going from one temperature to another. Unlike pressure-volume work done on the system, pressure-volume work done on the system absorbs heat without raising its temperature. This is because pressure-volume work on the system raises its temperature by a mechanism other than heating. It is because of this temperature dependence that a calorie is formally defined as the energy required to heat 1 g of water from 14.5 to 15.5 degrees Celsius instead of generally by just 1 degree Celsius.
Therefore, different methods can be used to determine heat capacity, most commonly at constant pressure and volume. To indicate the meaning of the measured value, the subscripts (p and V, respectively) are usually used. Typically, gas and liquid measurements are also based on constant volume. As the temperature increases, the substance expands against the constant pressure as it is measured at constant pressure. Therefore, the constant pressure measurements are greater than those at constant volume. In gases, significantly greater values are typically found under constant pressure than under constant volume, especially for gases at constant pressure.
Molar Specific Heat Capacity at Constant Pressure: If the heat transfer to the sample takes place at the same pressure as the sample remains, this method is known as Molar Specific Heat Capacity at Constant Pressure.
Molar Specific Heat Capacity at Constant Volume: If the sample is converted to heat by keeping its volume constant, the actual heat produced by this process is known as Molar Specific Heat Capacity at Constant Volume.
Points to Consider
Mass and volume are irrelevant for the specific heat capacity as opposed to the total heat capacity. In order to raise the temperature of a given substance by one degree Celsius, the amount of heat it takes to warm that mass by one degree Celsius. Special heat capacity is measured in J/(kg °C) or equivalently in J/(kg K).
C=cm or c=C/m is the relationship between the capacity for heat and the specific heat.
The mass m, specific heat c, change in temperature ΔT, and heat added (or subtracted) Q are related by the equation: Q=mc Temperature and phase of substances have an effect on specific heat values. Since they are difficult to calculate, they are measured empirically and available in tables as references.
Key Terms
In thermodynamics, specific heat capacity can be defined as the amount of heat needed to raise or lower the temperature of a unit mass of a substance by one degree Celsius. Specifically, it is a property of intensities.
FAQs on Specific Heat at Constant Pressure and Constant Volume
1. What is meant by the specific heat of a gas at constant volume (C_v)?
The specific heat of a gas at constant volume (C_v) is defined as the amount of heat energy required to raise the temperature of a unit mass of that gas by one degree (Kelvin or Celsius), while keeping its volume constant. Since the volume does not change, the gas does no external work (W=0). Therefore, according to the First Law of Thermodynamics, all the heat supplied is used exclusively to increase the internal energy of the gas.
2. What is meant by the specific heat of a gas at constant pressure (C_p)?
The specific heat of a gas at constant pressure (C_p) is the amount of heat energy needed to raise the temperature of a unit mass of that gas by one degree, while keeping its pressure constant. In this process, the gas is allowed to expand. The supplied heat is used for two purposes: first, to increase the internal energy of the gas (raising its temperature), and second, to do work on the surroundings as it expands.
3. Why is the specific heat at constant pressure (C_p) always greater than at constant volume (C_v)?
C_p is always greater than C_v because at constant pressure, the supplied heat energy has to perform two tasks: increase the internal energy of the gas and do external work as the gas expands against the constant pressure. In contrast, at constant volume, all the heat supplied is used only to increase the internal energy. To achieve the same one-degree rise in temperature, more heat must be supplied at constant pressure to account for the energy lost as work, making C_p greater than C_v.
4. What is Mayer's formula, and what relationship does it establish between C_p and C_v?
Mayer's formula describes the relationship between the two principal specific heats for an ideal gas. The formula is C_p - C_v = R, where R is the Universal Gas Constant. This equation signifies that the difference between the specific heat at constant pressure and constant volume is exactly equal to the amount of work done by one mole of the gas when its temperature is raised by one Kelvin at constant pressure.
5. What is the physical significance of the gas constant 'R' in Mayer's relation?
In the context of Mayer's relation (C_p - C_v = R), the gas constant 'R' represents the work done by one mole of an ideal gas when its temperature increases by one Kelvin (1 K) under constant pressure. It quantifies the portion of heat energy that is converted into mechanical work during isobaric (constant pressure) expansion, which is precisely the reason why C_p is larger than C_v.
6. How does the ratio of specific heats (γ = C_p/C_v) relate to the atomicity of a gas?
The ratio of specific heats, known as the adiabatic index (γ), is directly dependent on the degrees of freedom of the gas molecules, which is determined by its atomicity. Its value indicates how energy is stored in a gas. The typical values for ideal gases are:
- Monatomic gases (e.g., Helium, Argon): γ ≈ 1.67
- Diatomic gases (e.g., Oxygen, Nitrogen): γ ≈ 1.40
- Polyatomic gases (e.g., Carbon Dioxide, Methane): γ ≈ 1.33
7. Which thermodynamic process is more efficient for raising a gas's temperature: heating at constant volume or constant pressure?
Heating a gas at constant volume is more efficient for raising its temperature. In this process, all the heat supplied is converted directly into increasing the gas's internal energy, which results in a temperature rise. In contrast, during heating at constant pressure, a part of the supplied heat energy is used by the gas to do work as it expands, so not all the energy contributes to increasing the temperature. Therefore, for the same amount of heat supplied, the temperature increase will be greater at constant volume.
8. What is the fundamental principle behind the derivation of Mayer's relation (C_p - C_v = R)?
The derivation of Mayer's relation is fundamentally based on the First Law of Thermodynamics (ΔQ = ΔU + ΔW) applied to one mole of an ideal gas under two different conditions:
- At constant volume, no work is done (ΔW=0), so the heat supplied is equal to the change in internal energy: ΔQ = ΔU = C_vΔT.
- At constant pressure, the heat supplied is ΔQ = C_pΔT. The work done is ΔW = PΔV.
