Chemistry Q126 – AP-EAMCET 2025

Home Chemistry ThermodynamicsQuestion #627

For one mole of an ideal gas, three isochores were obtained at

V_{1}

V_{2}

and

V_{3}

separately. Their slopes are

m_{1}

m_{2}

and

m_{3}

, respectively. If

V_{1} < V_{2} < V_{3}

, then the correct relationship of slopes is

Clue 1 of 3: Ideal Gas Law

Recall the ideal gas law and how it relates pressure, volume, temperature, and the number of moles.

🧠 Full Solution Path

Step 1: Recall the Ideal Gas Law and define an Isochore✦ Active

The ideal gas law is given by $P V = n R T$ , where $P$ is pressure, $V$ is volume, $n$ is the number of moles, $R$ is the ideal gas constant, and $T$ is the absolute temperature. An isochoric process is one where the volume ( $V$ ) of the gas is kept constant.

💡 Teacher's Secret Hint

Remember that $T$ must always be in Kelvin for the ideal gas law.

Step 2: Express Pressure as a function of Temperature for an Isochore○ Expand

For an isochoric process, $V$ is constant. Rearranging the ideal gas law to express pressure ( $P$ ) in terms of temperature ( $T$ ):

P = (\frac{n R}{V}) T

This equation represents a straight line passing through the origin on a $P - T$ graph, with $P$ on the y-axis and $T$ on the x-axis.

💡 Teacher's Secret Hint

This equation is analogous to $y = m x$ , where $m$ is the slope.

Step 3: Determine the slope and its dependence on Volume○ Expand

Comparing $P = (\frac{n R}{V}) T$ with the standard linear equation $y = m x$ , the slope ( $m_{s l o p e}$ ) of the $P - T$ graph for an isochoric process is given by:

m_{s l o p e} = \frac{n R}{V}

Since $n$ (number of moles) and $R$ (ideal gas constant) are constants, the slope is inversely proportional to the volume ( $V$ ). That is, $m_{s l o p e} \propto \frac{1}{V}$ .

Step 4: Apply the given condition to find the relationship between slopes○ Expand

We are given that $V_{1} < V_{2} < V_{3}$ . Since the slope is inversely proportional to the volume, a smaller volume corresponds to a larger slope. Therefore, if $V_{1} < V_{2} < V_{3}$ , then $\frac{1}{V_{1}} > \frac{1}{V_{2}} > \frac{1}{V_{3}}$ . This implies that the slopes will follow the relationship:

m_{1} > m_{2} > m_{3}

💡 Teacher's Secret Hint

Inverse proportionality means that as one quantity increases, the other decreases. Double-check the inequality direction when taking reciprocals.

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Chemistry Question 126 – AP-EAMCET 2025

Clue 1 of 3: Ideal Gas Law

🧠 Full Solution Path