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# The density of a material A is $1500$ $kg{m^{ - 3}}$ and that of another material B is $2000$ $kg{m^{ - 3}}$. It is found that the heat capacity of 8 volumes of A is equal to heat capacity if 12 volumes of B. The ratio of specific heats of A and B will beA. $1:2$B. $3:1$C. $3:2$D. $2:1$

Last updated date: 18th Jun 2024
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Hint:Consider the definitions of the heat capacity and specific heat capacity to derive a relation between them. Apply the given condition and find the ratio of specific heats of A and B.

Heat capacity: It is a measurable quantity that gives the amount of heat required to raise the temperature of a body by one degree Celsius. Now for heat capacity, heat supplied to a body/absorbed by a body is proportional to the corresponding change in temperature.
$\Delta Q \propto \Delta T \\ \Rightarrow\Delta Q = C\Delta T \\ \Rightarrow C = \dfrac{{\Delta Q}}{{\Delta T}} \\$
Here, $C$ is heat capacity.

Specific heat capacity: It is a measurable quantity that gives the amount of heat required to raise the temperature of the body of unit mass by one degree Celsius. For Specific heat capacity, heat supplied to a body/absorbed by a body is proportional to mass of the body and corresponding change in temperature.
$\Rightarrow \Delta Q \propto m \\ \Rightarrow\Delta Q \propto \Delta T \\ \Rightarrow \Delta Q \propto m\Delta T \\ \Rightarrow \Delta Q = ms\Delta T \\$
Here, $s$ is the specific heat.

Now, if you compare, the ratio of heat absorbed by the body and corresponding change in temperature is $\dfrac{{\Delta Q}}{{\Delta T}}$ which should be the same since they represent the same quantity. Hence you can equate both the values of the ratio obtained.
$\therefore C = ms$
Now, given that the heat capacity is the same for both the materials.
${m_A}{s_A} = {m_B}{s_B}$ and we have$m = \rho V$
$\Rightarrow {\rho _A}{V_A}{s_A} = {\rho _B}{V_B}{s_B} \\ \Rightarrow\dfrac{{{s_A}}}{{{s_B}}} = \dfrac{{{\rho _B}{V_B}}}{{{\rho _A}{V_A}}} \\ \Rightarrow\dfrac{{{s_A}}}{{{s_B}}} = \dfrac{{2000 \times 12}}{{1500 \times 8}} \\ \therefore\dfrac{{{s_A}}}{{{s_B}}} = 2 \\$
Therefore, the ratio of specific heats of A and B will be $2:1$.Hence, option (D) is correct.

Note: Always keep in mind the difference between heat capacity and specific heat capacity. Although in definition of both the quantities we used the unit of temperature as Celsius, in equation we use the unit of temperature as Kelvin.