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MCQEasyJEE 2024Dimensions & Dimensional Analysis

JEE Physics 2024 Question with Solution

The equation of state of a real gas is given by (P+aV2)(V−b)=RT\left( P + \frac{a}{V^2} \right)(V - b) = RT, where PP, VV, and TT are pressure, volume, and temperature. The dimensions of ab2\frac{a}{b^2} are similar to:

  • A

    PP

  • B

    PVPV

  • C

    RTRT

  • D

    RR

Answer

Correct answer:A

Step-by-step solution

Standard Method

Given: The equation of state is

(P+aV2)(V−b)=RT\left( P + \frac{a}{V^2} \right)(V - b) = RT

Find: The dimensions of ab2\frac{a}{b^2}.

Since PP and aV2\frac{a}{V^2} are added, they must have the same dimensions. Therefore,

[aV2]=[P]\left[ \frac{a}{V^2} \right] = [P]

Now volume has dimensions

[V]=[L3][V] = [L^3]

so

[V2]=[L6][V^2] = [L^6]

Hence,

[a]=[P][V2]=[ML−1T−2][L6]=[ML5T−2][a] = [P][V^2] = [M L^{-1} T^{-2}][L^6] = [M L^5 T^{-2}]

Also, since V−bV - b appears in the equation, bb must have the same dimensions as volume:

[b]=[L3][b] = [L^3]

Therefore,

[b2]=[L6][b^2] = [L^6]

Now,

[ab2]=[a][b2]=[ML5T−2][L6]=[ML−1T−2]\left[ \frac{a}{b^2} \right] = \frac{[a]}{[b^2]} = \frac{[M L^5 T^{-2}]}{[L^6]} = [M L^{-1} T^{-2}]

This is the dimensional formula of pressure.

Therefore, the correct option is A and the dimensions of ab2\frac{a}{b^2} are the same as PP.

Dimension Matching Trick

Given:

(P+aV2)(V−b)=RT\left( P + \frac{a}{V^2} \right)(V - b) = RT

Find: The dimensions of ab2\frac{a}{b^2}.

Use direct matching. Since aV2\frac{a}{V^2} is added to PP,

[a]=[P][V2][a] = [P][V^2]

And since bb is subtracted from VV,

[b]=[V][b] = [V]

So,

[ab2]=[P][V2][V2]=[P]\left[ \frac{a}{b^2} \right] = \frac{[P][V^2]}{[V^2]} = [P]

Therefore, the correct option is A.

Common mistakes

  • Assuming bb is dimensionless is incorrect because V−bV - b is a subtraction, so both terms must have the same dimensions. Therefore, take [b]=[V]=[L3][b] = [V] = [L^3].

  • Using the answer key label from the page without checking the working is a mistake. The solution steps show that ab2\frac{a}{b^2} has the dimensions of pressure, so the correct option is the one containing PP.

  • Confusing [a][a] with pressure directly is wrong. From aV2\frac{a}{V^2} having dimensions of pressure, aa must equal pressure multiplied by volume squared.

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