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Properties of Matter question

2023 · 13 Apr · Shift 1 · Q56
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Properties of Matter question

2023 · 13 Apr · Shift 1 · Q56

JEE MainPhysicsProperties of MatterMCQ+4 / −1
Under isothermal condition, the pressure of a gas is given by P=a V−3\mathrm{P}=a \mathrm{~V}^{-3}P=a V−3, where aaa is a constant and V\mathrm{V}V is the volume of the gas. The bulk modulus at constant temperature is equal to
  1. A
    P2\frac{P}{2}2P​
  2. B
    2 P
  3. C
    3 P
  4. D
    P
View written solutionFree

Correct answer: C

  1. Bulk modulus definition

    The isothermal bulk modulus is BT=−VdPdVB_T=-V\frac{dP}{dV}BT​=−VdVdP​

  2. Given equation of state

    P=aV−3P=aV^{-3}P=aV−3

    Differentiate with respect to VVV: dPdV=a(−3)V−4=−3aV−4\frac{dP}{dV}=a(-3)V^{-4}=-3aV^{-4}dVdP​=a(−3)V−4=−3aV−4

  3. Substitute into bulk modulus formula

    BT=−V(−3aV−4)=3aV−3B_T=-V\left(-3aV^{-4}\right)=3aV^{-3}BT​=−V(−3aV−4)=3aV−3

  4. Use P=aV−3P=aV^{-3}P=aV−3

    Since aV−3=PaV^{-3}=PaV−3=P therefore, BT=3PB_T=3PBT​=3P

  5. Check options

    • A: P2\frac{P}{2}2P​ ❌
    • B: 2P2P2P ❌
    • C: 3P3P3P ✅
    • D: PPP ❌

Hence, the correct answer is Option C.

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