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Mechanical Properties of Solids

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Question : 12 of 21
Marks: +1, -0
Compute the bulk modulus of water from the following data : Initial volume = 100.0 litre, Pressure increase = 100.0 atm (1 atm = 1.013 × 105^{5}Pa), Final volume = 100.5 litre. Compare the bulk modulus of water with that of air (at constant temperature). Explain in simple terms why the ratio is so large.
Solution:  
p = 100 atmosphere = 100×1.013×105100 \times 1.013 \times 105Pa
Initial volume,V1=100 litre=100×103 m3V_1 = 100 \text{ litre} = 100 \times 10^{-3} \text{ m}^3
Final volume, V2=100.5 litre=100.5×103 m3V_2 = 100.5 \text{ litre} = 100.5 \times 10^{-3} \text{ m}^3
ΔV=\therefore \Delta V = change in volume =V2V1= V_2 - V_1
=(100.5100)×103 m3= (100.5 - 100) \times 10^{-3} \text{ m}^3
=0.5×103 m3= 0.5 \times 10^{-3} \text{ m}^3
Bw=B_w = bulk modulus of water =?= ?
\therefore From formula, Bw=PΔVV,B_w = \frac{P}{\frac{\Delta V}{V}}, we get
Btw=PVΔVB_{tw} = \frac{P V}{\Delta V}
=100×1.013×105×100×1030.5×103(V=V1)= \frac{100 \times 1.013 \times 10^{5} \times 100 \times 10^{-3}}{0.5 \times 10^{-3}} (\because V = V_1)
or Bw=2.026×109 PaB_w = 2.026 \times 10^{9} \text{ Pa}
Also we know that the bulk modulus of air at S.T.P is given by
Bair=105 PaB_{\text{air}} = 10^{5} \text{ Pa}
BwBair=2.026×109105\therefore \frac{B_w}{B_{\text{air}}} = \frac{2.026 \times 10^{9}}{10^{5}}
=2.026×104=20260= 2.026 \times 10^{4} = 20260
The ratio is too large. This is due to the fact that the strain for air is much larger than for water at the same temperature. In other words the intermolecular distances in case of liquids are very small as compared to the corresponding distances in the case of gases. Hence there are larger interatomic forces in liquids than in gases.
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