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Mechanical Properties Of Fluids

Question
CBSEENPH11019360
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The coefficient of volume expansion of glycerin is 49 x l0–5 per °C. What is the fractional change in its density for 30°C rise in temperature?

Solution
Let m be the mass of glycerin occupying the volume V when its density is d.
When we increase the temperature, the volume of same mass increases (let it be V') and hence density decreases.
Let it be ρ'. 
Therefore,
             space space space space space ρV space equals space straight rho apostrophe straight V apostrophe 
                 straight rho apostrophe space equals space straight rho fraction numerator straight V over denominator straight V apostrophe end fraction  
If straight theta be the increase in temperature and straight gamma  be coefficient of volume expansion then, 
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∴                 straight rho apostrophe space equals space straight rho fraction numerator straight V over denominator straight V left parenthesis 1 plus γθ right parenthesis end fraction space equals space straight rho left parenthesis 1 minus γθ right parenthesis 
∴                 straight rho apostrophe space equals space straight rho minus pγθ 
Now, fractional change in density is, 
            fraction numerator straight rho apostrophe minus straight rho over denominator straight rho end fraction equals negative γθ 
Negative sign shows that density decreases.
Therefore fractional decrease in density is, 
γθ space equals space 49 cross times 10 to the power of negative 5 end exponent cross times 30 space equals space 0.0147,  for 30o C rise in temperature. 

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