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Isothermal process is a process in which the temperature of the system remains constant during the change from its initial to the final stage. In this process, the heat is transferred in or out of the system gradually such that the equilibrium is maintained. It is a thermodynamic process. Thermodynamics is the branch of physics that deals with the relation between heat, work done, temperature and energy.
In an isothermal process, the change in temperature is zero i.e. \(\triangle\)T = 0
What is Isothermal Process?
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A thermodynamic process in which the temperature of the system, undergoing a change, remains constant is known as an isothermal process. The system changes heat with the surrounding in a way that the thermal equilibrium of the system is maintained. For an isothermal process,
dT = 0
However, the other factors such as volume and pressure may vary. According to Boyle’s Law, the ideal gas equation for the isothermal process is given by;
PV = Constant
This suggests that the pressure and volume of a particular substance are inversely proportional to one another in the case of isothermal process.
P-V Graph for an Isothermal Process
Isothermal Process for an Ideal GasLet us suppose that an ideal gas at constant temperature changes from its initial state to the final state with variations in volume and temperature. The gas absorbs heat and does work in the case of isothermal expansion, but the work is done on the gas as well by the environment. During this process, the heat is released. So, to maintain a constant temperature, either the heat is transferred into the system or shifted out of the system. If the work is done on the gas, internal energy may vary leading to a change in temperature. In other words, the work should either be done on the system or by the system. When referred to as an ideal gas, its internal energy solely depends on temperature. In such cases, there is no change in internal energy. |
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Examples of Isothermal Process
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We can observe many examples of the isothermal processes in our day-to-day life. Since the temperature is constant and there is only a change in volume and pressure of the substances, the isothermal process can be considered as a reversible process. A few examples of isothermal process are given below:
- Change in the phase or state (solid, liquid, gas) of substances due to melting or evaporation.
- Carnot cycle works at a constant temperature.
- The reactions taking place in the refrigerator are performed at constant temperature and are considered isothermal.
- Through the isothermal process, interactions between a cell and the surrounding cells take place in our body.
- The heat pump that either transfers heat into the house or shifts the heat outside from the house, is also an example of isothermal process.
Boiling of Water is an example of Isothermal Process
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Boyle’s Law
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An ideal gas can be understood as a hypothetical gas whose molecules face an elastic collision and do not interact with one another. Now, according to Joule’s second law, the internal energy of a fixed volume of an ideal gas depends only on its temperature. Hence, for an isothermal process, the internal energy of an ideal gas should be constant. This is known as Boyle’s Law.
Boyle’s LawBoyle’s Law states that the product of the volume and pressure for an ideal gas remains constant. In other words, in case of an ideal gas, the pressure exerted by an object varies inversely with the volume occupied by it if the temperature and amount of the gas remains constant. |
Boyle’s Law Equation
According to Boyle’s law, for an ideal gas, the pressure of the given mass is inversely proportional to its volume when the temperature is constant. It implies that,
PV = k
Here, k is a constant, P is the pressure and V is the volume.
From this law, we can derive,
P1V1 = P2V2
where P1 and V1 are the pressure and volume of the ideal gas at the initial state and P2 and V2 are the pressure and volume of the ideal gas at the final state.
It can be inferred that when the volume doubles, the pressure is reduced by half and vice versa. To keep the equations balanced a new constant was introduced in the equation, making the equation as;
PV = nRT
Here,
- P is the pressure
- V is the volume
- n is the no. of moles
- R is the universal gas constant
- T is the temperature
First Law of Thermodynamics for Isothermal Process
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The First Law of Thermodynamics states that the change in internal energy, \(\triangle\)U, of a system is equal to the heat that is added to the system (Q) minus the work done (W) by the system.
\(\triangle\)U = Q – W
In an Isothermal process, the increase in internal energy leads to an increase in temperature. This means that the internal energy is directly proportional to the temperature. The general expression used for the internal energy of an ideal gas is,
U = 3/2 nRT
From the above equation, it can be inferred that constant internal energy is maintained at constant temperature and \(\triangle\)U = 0. So, the First Law of Thermodynamics can be rearranged as follows:
Q = W
This implies that heat added to the system is equal to the amount of work done by the system, or in other words work is done by the heat added to the system. So, either the work is done by the system or on the system. Hence, it can be concluded that in an isothermal process the temperature remains constant.
Difference Between Isothermal and Adiabatic Process
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An adiabatic process is a type of thermodynamic process in which no heat transfer occurs between the system and the surroundings. However, it does not implies that the temperature is constant in the process. It only refers that neither heat is transferred into the system nor out of the system. It is a reversible process.
Whereas, isothermal process is a thermodynamic process in which the temperature of the system remains constant. There is a transfer of heat occurring in the system, but it happens too slow so that the thermal equilibrium is always maintained.
P-V Graph for Isothermal and Adiabatic Process
The main differences between the two processes are discussed below:
| Parameter | Isothermal Process | Adiabatic Process |
|---|---|---|
| Definition | The thermodynamic process that occurs at a constant temperature is called as isothermal process. | The thermodynamic process in which no heat is transferred between the system and its surrounding is called adiabatic process. |
| Pressure and Volume | When compared the pressure is more than the volume. | When compared the volume is more than the pressure. |
| Transfer of Heat | Heat transfer takes place between the system and its surroundings. | No heat transfer takes place. |
| Heat | Here, heat is either added to or removed from the system to maintain a constant temperature. | Neither heat is transferred into the system nor out of the system. |
| Temperature | Since the heat transfer occurs, the temperature is maintained as a constant. | Due to variations in the internal system of the process, the temperature varies. |
| Rate of Transformation | The transformation is observed to occur at a slow pace. | The transformation occurs at a fast rate. |
| Examples | Change of state of the matter due to melting, evaporation, and cooling processes, carnot engine, refrigerator, heat pump used in houses. | Pneumatic tire, oscillation of pendulum in a vertical plane, quantum harmonic oscillator, vertical flow of air in the atmosphere. |
Things to Remember
- An isothermal process is a thermodynamic process that occurs at a constant temperature with heat transfer between the system and its surroundings.
- The condition for an isothermal process is dT = 0.
- Changes in the phase of matter, carnot cycle, refrigerator, heat pump are some of the examples of the isothermal processes.
- Boyle’s Law equation for an ideal gas in the isothermal process can be given by the formula PV = nRT.
- From the first law of thermodynamics, the general formula used for the internal energy of an ideal gas is U = 3/2 nRT.
- In a adiabatic process, there is no heat transfer between the system and the surrounding.
Sample Questions
Ques. A what temperature the adiabatic change is equivalent to the isothermal change? (1 Mark)
Ans. At 0 Kelvin temperature, the adiabatic change will be equivalent to the isothermal change.
Ques. What all changes occur in a thermodynamic system? (1 Mark)
Ans. A thermodynamic system undergoes changes in pressure, volume and internal energy.
Ques. Under what conditions the heat evolved or absorbed is equal to the internal energy change? (2 Marks)
Ans. At constant volume, the heat evolved or absorbed is equal to the internal energy change.
Ques. Name two intensive and extensive properties of a system. (2 Marks)
Ans. The intensive properties of a system includes viscosity, refractive index and extensive properties include mass, volume, heat capacity etc.
Ques. Calculate the maximum work obtained when 0.75 mol of an ideal gas expands isothermally and reversible at 27°C from a volume of 15 L to 25 L. (3 Marks)
Ans. For isothermal reversible expansion of an ideal gas,
W = – nRT log (V2/V1)
W = – 2.303 nRT log (V2/V1)
W = – 2.303 × 0.75 × 8.314 × 300 log (25/15)
W = -955.5 J
Ques. What are the properties of an isothermal process? (3 Marks)
Ans. The thermodynamic process that occurs at a constant temperature with the transfer of heat between the system and its surroundings is called as isothermal process. The properties of the isothermal process are:
- The temperature of the system remains constant.
- The equation is PV = constant.
- The work done by n moles of an ideal gas in an isothermal expansion from volume V1 to V2 at temperature T is given by W = nRT ln(V2/V1)
- The change in internal energy, i.e. ΔU is zero.
- The heat supplied to the gas is equal to the work done by the gas.
Ques. What is the first law of thermodynamics? What are its applications in regard to isothermal processes? (5 Marks)
Ans. The first law of thermodynamics, often regarded as the law of conservation of energy, states that if a system absorbs heat dQ and the internal energy of the system changes while the system does work, dW, then
dQ = W + dU
W = P.dV.dQ
W = dU + PdV
According to Boyle’s Law, PV = constant
There is an exchange of heat between the system and its surroundings. The system should be compressed or expanded very slowly so that there is sufficient time for the exchange of heat to keep the temperature constant. Therefore, for an isothermal process, the first law of thermodynamics can be represented as:
dQ = 0 + PdV
dQ = PdV
When a gas undergoes isothermal compression, dV and hence PdV is negative and likewise, dQ will also become negative. Therefore, if we have to compress a gas isothermally, then an amount of heat, that is equivalent to the work done on the gas, will have to be removed.
Ques. A 0.5 mole of gas expands isothermally, at a temperature of 300 K, from an initial volume of 2 L to a final 6 L. Calculate the: (5 Marks)
a) Work done by the gas
b) Added heat to the gas
c) Final pressure of the gas
It is given that the value of the gas constant, R = 8.31 Jmol-1K-1
Ans. a). Given µ = 0.5
W = 0.5 mol × 8.31J/ mol.K × 300K ln(6L/2L)
W = 1.369 kJ
As the work done is by the gas, we get W as positive.
b). From the first law of thermodynamics, we infer that in an isothermal process the heat supplied to a system is spent to do work.
Hence, Q = W = 1.369 kJ.
Therefore Q is also positive. This implies that heat flows into the system.
c). For an isothermal process,
PiVi = PfVf = µRT
Pf = µRT/Vf = 0.5 mol × 8.31 J/mol.K × 300 K / (6 ×10-3m3)
= 207.75 k Pa
Ques. Calculate the amount of heat added to an ideal gas, undergoing an isothermal process, so that the gas does a work of 5000 Joules on its environment. (5 Marks)
Ans. Given, Work (W) = 5000 Joules
An isothermal process is one with a constant temperature
ΔU = 3/2 n R ΔT
Here, ΔU refers to the change in the internal energy,
N refers to the number of moles,
R is the universal gas constant,
And ΔT is the change in temperature.
Now, if ΔT = 0, then according to the above equation ΔU also becomes zero. From the first law of thermodynamics:
ΔU = Q – W
0 = Q – W
Q = W
Hence, Q = 5000 Joules
Ques. What is true for an isothermal process? (5 Marks)
- ΔT > 0
- ΔU = 0
- ΔQ = ΔW
- PV = constant
Ans. In an Isothermal Process,
The temperature remains constant, i.e. ΔT = 0
Since internal energy depends on the temperature:
ΔU = 0
From the first law of thermodynamics,
ΔU = ΔQ - ΔW
Since ΔU = 0
ΔQ = ΔW
Also, PV = nRT
As T is constant, PV = constant
Ques. An ideal gas with an initial pressure of 1.4 × 105 Pa and an initial volume of 0.50 m3 expands isothermally to a volume of 2.2 m3. Calculate the total amount of thermal energy that is transferred to the gas during this process? (5 Marks)
Ans. The ideal gas law states that:
PV = nRT
In the case of isothermal processes, the work done on the gas is:
W = nRT ln(Vf/Vo)
Now, since we are not given the number of moles and the temperature, we can substitute the ideal gas law such that:
W = PoVo ln(Vf/Vo)
Hence, the energy transferred during the process is:
W =(1.4×105) (0.50) ln(2.2/0.5)
= 1.04 × 105
Ques. Differentiate between isothermal and adiabatic process? (5 Marks)
Ans. The key differences between isothermal and adiabatic process are as follows:
| Isothermal Process | Adiabatic Process |
|---|---|
| In the isothermal process, there is an exchange of heat between the system and its surrounding. | In the adiabatic process, there is no heat exchange between the system and its surrounding. |
| The temperature remains constant. | The temperature may increase. |
| Heat can be released or added to the system just to keep the temperature constant. | Heat is neither added nor released from the system. |
| Here, transformation is slow. | Here, transformation is fast. |
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