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Properties of Matter question

2023 · 1 Feb · Shift 1 · Q71
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Properties of Matter question

2023 · 1 Feb · Shift 1 · Q71

JEE MainPhysicsProperties of MatterNumerical+4 / −1
A certain pressure 'P\mathrm{P}P' is applied to 1 litre of water and 2 litre of a liquid separately. Water gets compressed to 0.01%0.01 \%0.01% whereas the liquid gets compressed to 0.03%0.03 \%0.03%. The ratio of Bulk modulus of water to that of the liquid is 3x\frac{3}{x}x3​. The value of xxx is ‾\underline{\hspace{2cm}}​.
Numerical answer
View written solutionFree

Correct answer: 1

  1. Use the definition of bulk modulus

The bulk modulus is given by

B=PΔV/VB = \frac{P}{\Delta V / V}B=ΔV/VP​

where:

  • PPP = applied pressure
  • VVV = original volume
  • ΔV\Delta VΔV = decrease in volume

Since the same pressure PPP is applied to both liquids,

B∝1ΔV/VB \propto \frac{1}{\Delta V/V}B∝ΔV/V1​
  1. Find fractional compression for each
  • For water, compression is 0.01%0.01\%0.01%
0.01%=0.01100=10−40.01\% = \frac{0.01}{100} = 10^{-4}0.01%=1000.01​=10−4

So,

(ΔVV)water=10−4\left(\frac{\Delta V}{V}\right)_{\text{water}} = 10^{-4}(VΔV​)water​=10−4
  • For the liquid, compression is 0.03%0.03\%0.03%
0.03%=0.03100=3×10−40.03\% = \frac{0.03}{100} = 3 \times 10^{-4}0.03%=1000.03​=3×10−4

So,

(ΔVV)liquid=3×10−4\left(\frac{\Delta V}{V}\right)_{\text{liquid}} = 3 \times 10^{-4}(VΔV​)liquid​=3×10−4
  1. Take ratio of bulk moduli
BwaterBliquid=P/(10−4)P/(3×10−4)=3×10−410−4=3\frac{B_{\text{water}}}{B_{\text{liquid}}} = \frac{P/(10^{-4})}{P/(3\times 10^{-4})} = \frac{3\times 10^{-4}}{10^{-4}} = 3Bliquid​Bwater​​=P/(3×10−4)P/(10−4)​=10−43×10−4​=3

Thus,

BwaterBliquid=3\frac{B_{\text{water}}}{B_{\text{liquid}}} = 3Bliquid​Bwater​​=3
  1. Compare with given form

Given,

BwaterBliquid=3x\frac{B_{\text{water}}}{B_{\text{liquid}}} = \frac{3}{x}Bliquid​Bwater​​=x3​

So,

3x=3\frac{3}{x} = 3x3​=3

Hence,

x=1x = 1x=1
  1. Final answer
1\boxed{1}1​

The stored correct answer is also 111, so it matches.

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