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# First Law of Thermodynamics – Definition, Applications

First Law of Thermodynamics: The First Law of Thermodynamics is a fundamental rule that relates internal energy and work done by a system to the heat supplied to it. This law has played a very significant role in some of the greatest inventions like heat engines, refrigerators, air conditioners etc. Thermodynamics is the branch of physics that deals with the study of the transfer of heat between two bodies or system of bodies and the resulting change that takes place in internal energy and work done.

In this article we will state first law of thermodynamics, and discuss on what is thermodynamics and limitations of first law of thermodynamics in Chemistry. Read on to understand.

## First Law of Thermodynamics

First Law of thermodynamics is an extension of Law of Conservation of Energy, and it states that for a closed system, change in internal energy is equal to the difference of the heat supplied to the system and the work done by the system.

In other words, the heat supplied to a system is equal to the sum of the change in the internal energy of the system and work done by the system.

$$∆ Q – ∆ W = ∆ U$$
Where,
$$∆ Q$$ is the amount of heat supplied to the system.
$$∆ W$$ is the work done by the system.
$$∆ U$$ is the change in the internal energy of the system.
Thus, the heat supplied to the system is given by,
$$∆ Q = ∆ W + ∆ U$$

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1. Open system allows both the transfer of energy and mass.
2. Closed system only allows the transfer of energy and not mass.
3. Isolated system doesn’t allow the transfer of both mass and energy.

### Sign Convention

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### Heat

Heat is a form of energy that gives us the sensation of hotness or coldness of a body. For example, during the winter season, our surroundings are colder than our bodies, and due to that, heat is transferred from our bodies to the surroundings. It is because of this loss of heat that we feel cold. During the summer, the surroundings are hotter than our body, and thus, heat flows from the surroundings to our body and due to this heat gained, we feel hot.
1. Temperature change
2. Phase change
It is interesting to note that we cannot determine the amount of the heat of a substance or the total heat content of the substance, but we can only determine the heat lost or heat gained, i.e., we can only measure the amount of heat transfer.

### Total Internal Energy and Degrees of Freedom

Total Internal Energy: It is the property of a substance that signifies the total energy possessed by the constituent particles of that substance.
For gases, the internal energy consists of,

1. Kinetic Energy
a. Translational Kinetic Energy
b. Rotational Kinetic Energy
c. Vibrational Kinetic Energy (Significant only at higher temperatures)
2. Potential energy
a. Intermolecular Potential Energy (it is zero for ideal gases)
b. Intramolecular Potential Energy (Significant only at higher temperatures)

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Degrees of Freedom: It is the total number of possible independent motions for a given molecule. According to the Law of Equipartition of Energy, the total energy of a molecule is divided equally among all its degrees of freedom.

Total internal energy of an ideal gas for its one mole and for its one degree of freedom is given by,
$$\frac{1}{2}RT$$
$$T$$ is the absolute temperature.
Thus, the total internal energy of the gas is given by,
$$U = \frac{1}{2}nfRT$$
Change in internal energy is given by,
$$\Delta U = \frac{1}{2}nfR \Delta T$$
$$n$$ is the number of moles of the gas.
$$f$$ is the total number of degrees of freedom.
$$R$$ is the universal gas constant.
$$T$$ is the absolute temperature.
Therefore, we can infer that the change in internal energy is a state function i.e., it only depends on the final and the initial state of the gas.

### Work

Work: When heat is supplied to a system, a portion of this heat can be utilised by the system to do some work. for example, if we take a gas-piston system, where some gas is filled in a cylinder fitted with a movable piston, then supplying heat to the system can cause an expansion or compression of the gas and the movement of the piston. In other words, there is some work done. Let us try to derive an expression for this work done.

Let $$F$$ be the force acting on the piston due to the gas and $$dx$$ be the small displacement of the piston, then
$$dW = F \cdot dx = \frac{F}{A}Adx$$
$$P = \frac{F}{A}$$
$$dV = Adx$$
$$\Rightarrow dW = P \cdot dV$$
$$\Delta W = \int_{{V_i}}^{{V_f}} {PdV}$$
Where,
$$dV$$ is the infinitesimal change in volume.
$$P$$ is the pressure at that instance.
We can see that in order to calculate the work done by a gas, we have to add all the infinitesimal work that the gas performs while going through the process. This means we have to know the pressure at every instant throughout the process. Therefore, we can infer that the work done is a path function, i.e., it depends on the path or the process between the final and the initial states.

### Heat Capacity

The Heat Capacity of a system is defined as the amount of heat required to change the temperature of the system by $$1\,\rm{K}.$$
$$C = \frac{{\Delta Q}}{{\Delta T}}$$
Where,
$$∆ Q$$ is the amount of heat transferred.
$$∆ T$$ is the change in temperature.
The amount of heat transferred per unit change in temperature per mole of the substance is known as molar heat capacity.
$$C = \frac{{\Delta Q}}{{n\Delta T}}$$
Where $$n$$ is the number of moles of the substance.

### Thermodynamic Process

The thermodynamic state of gas-piston system is defined by four parameters- Pressure, Volume, Temperature, and Number of Moles. Since these four are related by the Ideal Gas Equation, therefore, effectively, we need any three parameters to define the state of the system. As the system undergoes a process, the value of these parameters and hence the state of the system changes.

We can plot a graph between any two parameters of the system in a two-dimensional plane undergoing a process. The curve which we have obtained is representing a thermodynamic process. This graphical representation of a thermodynamic process is known as an indicator diagram. The equation of the curve is known as the equation of the thermodynamic process.

### Isothermal Process

1. $$T =$$ const
2. $$∆ U = 0 ⇒ U =$$ const
3. $$P V =$$ const (Equation of the process)
$${P_i}{V_i} = {P_f}{V_f}$$
4. $${{\Delta }}W = \int_{{V_i}}^{{V_f}} {\frac{{nRT}}{V}dV}$$
$${{\Delta }}W = nRT\ln \left({\frac{{{V_f}}}{{{V_i}}}} \right)$$
$${{\Delta }}W = nRT\ln \left({\frac{{{P_i}}}{{{P_f}}}} \right)$$
5. $$C_{iso} = \infty$$

Indicator Diagram

### Isobaric Process

1. $$P =$$ const
2. $$∆ U = nC_{v} ∆ T$$
3. $$\frac{V}{T} =$$const
$$\frac{{{V_i}}}{{{T_i}}} = \frac{{{V_f}}}{{{T_f}}}$$
4. $$\Delta W = \mathop \smallint \nolimits_{{V_i}}^{{V_f}} PdV = P\left({{V_f} – {V_i}} \right) = P{{\Delta }}V$$
$$∆ W = P ∆ V$$
$$∆ W = nR ∆ T$$
5. $${C_p} = \frac{{{{\left({\Delta Q} \right)}_p}}}{{n\Delta T}} =$$ const

Indicator Diagram

### Isochoric Process

1. $$V =$$ const
2. $$∆ U = nC_{v} ∆ T$$
3. $$\frac{P}{T} =$$ const
$$\frac{{{P_i}}}{{{T_i}}} = \frac{{{P_f}}}{{{T_f}}}$$
4. $$\Delta W = \int_{{V_i}}^{{V_f}} {PdV = P\left({{V_f} – {V_i}} \right) = 0}$$
$$∆ W = 0$$
5. $${C_v} = \frac{{{{\left({\Delta Q} \right)}_v}}}{{n\Delta T}} =$$ const

Indicator Diagram

1. $$∆ Q =$$ const
2. $$∆ U = nC_{v} ∆ T$$
3. $$PV^{γ} =$$ const
$$TV^{γ-1} =$$ const
4. Work done is given by,
$$\Delta W = \int_{{V_i}}^{{V_f}} {PdV}$$
$$= \int_{{V_i}}^{{V_f}} {\frac{K}{{{V^\gamma }}}dV}$$ $$P{V^\gamma } = K$$
$$= K\left[{\frac{{{V^{ – \gamma + 1}}}}{{ – \gamma + 1}}} \right]_{{V_i}}^{{V_f}}$$
$$\frac{{KV_f^{ – \gamma + 1} – KV_i^{ – \gamma + 1}}}{{ – \gamma + 1}}$$
$$\Rightarrow \Delta W = \frac{{{P_f}{V_f} – {P_i}{V_i}}}{{1 – \gamma }}$$
$$\Rightarrow \Delta W =\frac{{nR\left({{T_i} – {T_f}} \right)}}{{1 – \gamma }}$$
5. $$C_{\rm{adiabatic}}=0$$

Indicator Diagram

### What is the Application of First Law of Thermodynamics?

The First Law of thermodynamics finds its application in the following areas.
Refrigerators
It’s interesting to know, if a refrigerator door is kept open inside a closed room, the temperature of the room will increase.

Thermal Power Plants

Engines

### First Law of Thermodynamics- Sample Problems

Q.1. Draw the approximate plots of adiabatic processes, for the case of an ideal gas, using the following variables,
$$T^{γ} P^{1-γ} =$$ const
$$T^{\frac{γ}{1-γ}} P =$$ const
As we know that,
$$γ > 1$$
$$⇒\frac{1-γ}{γ} < 0$$
Let,
$$\frac{{1 – γ}}{γ} = \, – \,B$$
$$⇒ T^{-B} P = \rm{const} = C$$
$$⇒ P = CT^B$$
$$0 < B < 1$$
The graph will be,

Q.2. Find the molar heat capacity of a polytropic process $$PV^n =$$ const.
Ans: Given,
$$PV^n =$$ const
From first law of thermodynamics we have,
$$∆ Q = ∆ U + W$$
Now, dividing the equation with, $$∆ T$$, we get,
$$\frac{{{{\Delta }}Q}}{{{{\Delta }}T}} = \frac{{{{\Delta }}U}}{{{{\Delta }}T}} + \frac{{{{\Delta }}W}}{{{{\Delta }}T}}$$
$$\frac{{{{\Delta }}Q}}{{{{\Delta }}T}} = \frac{{{C_v}{{\Delta }}T}}{{{{\Delta }}T}} + \frac{{PdV}}{{{{\Delta }}T}}$$
From ideal gas law,
$$dT = PdV + VdP$$
$$\Rightarrow C = {C_v} + \frac{{RPdV}}{{PdV + VdP}}$$
From the equation of the process,
$$P.nV^{n-1} dV + V^{n} dP = 0$$
$$dP =\frac {-nPdV}{V}$$
$$\Rightarrow C = {C_v} + \frac{{RPdV}}{{PdV + V\left({\frac{{ – nPdV}}{V}}\right)}}$$
Therefore, the molar heat capacity is given by,
$$\Rightarrow C = {C_v} + \frac{R}{{1 – n}}$$

### Summary

1. First law of thermodynamics states that for a closed system, change in internal energy is equal to the difference of the heat supplied to the system and the work done by the system.
2. The formula for first law of thermodynamics is given by,
$$∆ Q = ∆ W + ∆ U$$
3. Change in internal energy is given by,
$$∆ U = \frac{1}{2} nfR ∆ T$$
4. The work done by the system is given by,
$$\Delta W = \mathop \smallint \nolimits_{{V_i}}^{{V_f}} PdV$$

### First Law of Thermodynamics FAQs

Q.1. What does the first law of thermodynamics say?
Ans:
First law of thermodynamics states that for a closed system, change in internal energy is equal to the difference of the heat supplied to the system and the work done by the system.
∆𝑄=∆𝑊+∆𝑈

Q2. What are the three laws of Thermodynamics?
Ans.
The three laws of Thermodynamics are:

1. Zeroth Law of thermodynamics: This law states that if a thermodynamic system $$A$$ and system $$B$$ is in equilibrium with the third system $$C$$ then System $$A$$ and system $$B$$ will also be in equilibrium.
2. First Law of thermodynamics: This states that for a closed system, change in internal energy is equal to the difference of the heat supplied to the system and the work done by the system.
3. Second law of thermodynamics: This law states that no cyclic process can ever be $$100\%$$ efficient.
In other words, the entropy of the universe is always increasing.

Q.3. What is the 2nd law of thermodynamics in simple terms?
Ans: Second law of thermodynamic means that even if all the processes are ideal, any cyclic process cannot have $$100\%$$ efficiency.

Q4. What are some common applications of first law of Thermodynamics?
Ans.
Heat engine, refrigerators, air conditioners etc. are some of the common practical applications of first law of Thermodynamics

Q5. What is Degree of Freedom in Thermodynamics?
Ans.
The number of ways or directions in which a molecule can rotate, move or vibrate in space is known as Degree of Freedom in Thermodynamics.

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