A sample of 2 kg monoatomic helium (assumed ideal) is taken through the process ABC and another samp

Q: Why is the internal energy change the same for both paths?

Internal energy is a state function, meaning it only depends on the initial and final states (temperatures), not the path taken.

Q: How do we calculate the work done from the graph?

Work done is simply the area under the $p-V$ curve for the specific process. For an isobaric process, it's a rectangle's area.

Q: Why is work done zero for processes AB and DC?

These are isochoric processes, meaning the volume remains constant. Without a change in volume, no work is done by or on the gas.

Q: Can we determine the path just by looking at the final state?

No, the final state only gives us state variables like pressure, volume, and temperature. It doesn't contain information about the path taken to get there.

Q: Why do we use $C_V = \frac{3}{2}R$?

Helium is a monoatomic gas. For monoatomic gases, the molar heat capacity at constant volume, $C_V$, is $\frac{3}{2}R$ due to its 3 translational degrees of freedom.

Question

A sample of $2	ext{ kg}$ monoatomic helium (assumed ideal) is taken through the process $ABC$ and another sample of $2	ext{ kg}$ of the same gas is taken through the process $ADC$ (see fig). Given molecular mass of helium = $4$.
(a) What is the temperature of helium in each of the states $A, B, C$ and $D$ ?
(b) Is there any way of telling afterwards which sample of helium went through the process $ABC$ and which went through the process $ADC$? Write Yes or No.
(c) How much is the heat involved in the process $ABC$ and $ADC$ ?

Sigma Percentile · Accepted Answer

The problem presents us with a classic thermodynamic journey of an ideal monoatomic gas. We are given a $p-V$ graph showing two distinct paths, $ABC$ and $ADC$, taking the gas from an initial state $A$ to a final state $C$. Let's break down the physics step-by-step.

### Decoding the Gas State

First, we need to understand exactly how much gas we are dealing with. We are given $2	ext{ kg}$ of monoatomic helium.

To use the ideal gas law, we must convert this mass into moles. The molar mass of helium is $M = 4	ext{ g/mol}$.

$$n = \frac{m}{M} = \frac{2000	ext{ g}}{4	ext{ g/mol}} = 500	ext{ moles}$$

Now, we can determine the temperature at each state using the ideal gas equation, $pV = nRT$.

At state $A$, reading from the graph, the pressure $p_A = 5 	imes 10^4	ext{ N/m}^2$ and the volume $V_A = 10	ext{ m}^3$.

$$T_A = \frac{p_A V_A}{nR} = \frac{(5 	imes 10^4)(10)}{500 	imes 8.31} = 120.34	ext{ K}$$

Following the same logic for the other states, we observe how the macroscopic variables change. At state $B$, the pressure doubles while the volume remains constant, so the temperature must also double to $T_B = 240.68	ext{ K}$.

At state $C$, both pressure and volume are doubled compared to state $A$, leading to a temperature of $T_C = 481.36	ext{ K}$. Finally, at state $D$, the volume is doubled while the pressure is the same as at $A$, giving $T_D = 240.68	ext{ K}$.

### The State vs. Path Dilemma

Part (b) asks a fundamental conceptual question: Can we tell which path the gas took just by looking at the final state?

The answer is a resounding **No**.

Properties like pressure, volume, temperature, and internal energy are **state functions**. They depend exclusively on the current equilibrium state of the system, completely independent of the history or the path taken to reach that state.

### The First Law in Action

To find the heat involved in each process, we rely on the First Law of Thermodynamics: $Q = \Delta U + W$.

First, let's calculate the change in internal energy ($\Delta U$) from $A$ to $C$. Because internal energy is a state function, this value will be identical for both path $ABC$ and path $ADC$. For a monoatomic gas, the molar heat capacity at constant volume is $C_V = \frac{3}{2}R$.

$$\Delta U = nC_V(T_C - T_A) = 500 	imes \frac{3}{2} 	imes 8.31 	imes (481.36 - 120.34)$$

$$\Delta U = 2.25 	imes 10^6	ext{ J}$$

### Comparing the Two Paths

Now, let's evaluate the work done and heat for each specific path. Work done by a gas is geometrically represented by the area under the $p-V$ curve.

For path $ABC$, the process $AB$ is isochoric (constant volume), meaning no work is done. The process $BC$ is isobaric (constant pressure), so the work done is the area of the rectangle under $BC$.

$$W_{ABC} = p_B(V_C - V_B) = (10 	imes 10^4)(20 - 10) = 10^6	ext{ J}$$

Using the First Law, the total heat supplied along $ABC$ is:

$$Q_{ABC} = \Delta U + W_{ABC} = 2.25 	imes 10^6 + 10^6 = 3.25 	imes 10^6	ext{ J}$$

Conversely, for path $ADC$, the process $AD$ is isobaric and $DC$ is isochoric. The work done is the area under $AD$, which is noticeably smaller.

$$W_{ADC} = p_A(V_D - V_A) = (5 	imes 10^4)(20 - 10) = 0.5 	imes 10^6	ext{ J}$$

Again, applying the First Law:

$$Q_{ADC} = \Delta U + W_{ADC} = 2.25 	imes 10^6 + 0.5 	imes 10^6 = 2.75 	imes 10^6	ext{ J}$$

This beautifully demonstrates a core principle of thermodynamics: while the change in internal energy is path-independent, the work done and the heat exchanged are highly **path-dependent**.

Visualized Solution

Analyzing the $p - V$ Graph

Temperature at State $A$

Temperatures at $B, C,$ and $D$

State vs Path Functions

Change in Internal Energy ( $Δ U$ )

Work Done in Process $A B C$

Heat Involved in Process $A B C$

Work Done in Process $A D C$

Heat Involved in Process $A D C$

The Sigma Insight: First Law of Thermodynamics

Decoding the Gas State

The State vs

The First Law in Action

Comparing the Two Paths

Similar Questions

Comprehension Passage

(a) What is the temperature of helium in each of the states $A, B, C$ and $D$ ?

(b) Is there any way of telling afterwards which sample of helium went through the process $A B C$ and which went through the process $A D C$ ? Write Yes or No.

(c) How much is the heat involved in the process $A B C$ and $A D C$ ?

Comprehension Passage

If the process carried out on one mole of monatomic ideal gas is as shown in figure in the PV-diagram with $P_{0} V_{0} = \frac{1}{3} R T_{0}$ , the correct match is,

If the process on one mole of monatomic ideal gas is an shown is as shown in the TV-diagram with $P_{0} V_{0} = \frac{1}{3} R T_{0}$ , the correct match is

List-I describes thermodynamic processes in four different systems. List-II gives the magnitudes (either exactly or as a close approximation) of possible changes in the internal energy of the system due to the process.

List-I

List-II

A sample of an ideal gas is taken through the cyclic process abca as shown in the figure. The change in the internal energy of the gas along the path ca is $- 180 J$ . The gas absorbs $250 J$ of heat along the path ab and $60 J$ along the path bc. The work done by the gas along the path abc is

Following figure shows two processes A and B for a gas. If $Δ Q_{A}$ and $Δ Q_{B}$ are the amount of heat absorbed by the system in two cases, and $Δ U_{A}$ and $Δ U_{B}$ are changes in internal energies respectively, then

One mole of diatomic ideal gas undergoes a cyclic process ABC as shown in figure. The process BC is adiabatic. The temperatures at A, B and C are 400 K, 800 K and 600 K, respectively. Choose the correct statement.

One mole of a monatomic ideal gas is taken through a cycle $A B C D A$ as shown in the $p - V$ diagram. Column II gives the characteristics involved in the cycle. Match them with each of the processes given in Column I.

List-I

List-II

A reversible cyclic process for an ideal gas is shown below. Here, P , V and T are pressure , volume and temperature , respectively. The thermodynamic parameters q, w, H and U are heat, work, enthalpy and internal energy, respectively.

Visualized Solution

Analyzing the .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } p−V Graph

Temperature at State .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A

Temperatures at .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } B,C, and .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } D

State vs Path Functions

Change in Internal Energy ( .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ΔU)

Work Done in Process .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ABC

Heat Involved in Process .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ABC

Work Done in Process .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ADC

Heat Involved in Process .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ADC

The Sigma Insight: First Law of Thermodynamics

Decoding the Gas State

The State vs

The First Law in Action

Comparing the Two Paths

Similar Questions

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } (a) What is the temperature of helium in each of the states A,B,C and D?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } (b) Is there any way of telling afterwards which sample of helium went through the process ABC and which went through the process ADC? Write Yes or No.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } (c) How much is the heat involved in the process ABC and ADC?

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the process carried out on one mole of monatomic ideal gas is as shown in figure in the PV-diagram with P0​V0​=31​RT0​, the correct match is,

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the process on one mole of monatomic ideal gas is an shown is as shown in the TV-diagram with P0​V0​=31​RT0​, the correct match is

List-I describes thermodynamic processes in four different systems. List-II gives the magnitudes (either exactly or as a close approximation) of possible changes in the internal energy of the system due to the process.

List-I

List-II

One mole of diatomic ideal gas undergoes a cyclic process ABC as shown in figure. The process BC is adiabatic. The temperatures at A, B and C are 400 K, 800 K and 600 K, respectively. Choose the correct statement.

List-I

List-II

A reversible cyclic process for an ideal gas is shown below. Here, P , V and T are pressure , volume and temperature , respectively. The thermodynamic parameters q, w, H and U are heat, work, enthalpy and internal energy, respectively.

Analyzing the $p - V$ Graph

Temperature at State $A$

Temperatures at $B, C,$ and $D$

Change in Internal Energy ( $Δ U$ )

Work Done in Process $A B C$

Heat Involved in Process $A B C$

Work Done in Process $A D C$

Heat Involved in Process $A D C$

(a) What is the temperature of helium in each of the states $A, B, C$ and $D$ ?

(b) Is there any way of telling afterwards which sample of helium went through the process $A B C$ and which went through the process $A D C$ ? Write Yes or No.

(c) How much is the heat involved in the process $A B C$ and $A D C$ ?

If the process carried out on one mole of monatomic ideal gas is as shown in figure in the PV-diagram with $P_{0} V_{0} = \frac{1}{3} R T_{0}$ , the correct match is,

If the process on one mole of monatomic ideal gas is an shown is as shown in the TV-diagram with $P_{0} V_{0} = \frac{1}{3} R T_{0}$ , the correct match is