The linear coefficient of thermal expansion for water at 20 C is 69 times 10^(-6) K^(-1). What is...

Question:

The linear coefficient of thermal expansion for water at {eq}20 ^{\circ}C {/eq} is {eq}69 \times 10^{-6} K^{-1} {/eq}. What is the magnitude of the percentage change in density of {eq}1 \, m^3 {/eq} of water when it is heated from {eq}20 ^{\circ}C {/eq} to {eq}21 ^{\circ}C {/eq}? Density of water {eq}= 1000 \, kgm-3 {/eq}.

Coefficient of Expansion

The linear coefficient of thermal expansion is defined as the ratio of fractional change in length to the change in temperature. That is,

If the length of a rod changes from length {eq}l_i {/eq} to {eq}l_f {/eq} on the account of change in temperature {eq}\Delta T {/eq} then the linear coefficient of thermal expansion {eq}\alpha {/eq} is

{eq}\alpha = \dfrac{\frac{\Delta l}{l_i}}{\Delta T} = \dfrac{\frac{l_f - l_i}{l_i}}{\Delta T} {/eq}.

The volumetric coefficient of thermal expansion, similarly defined, can be expressed as

{eq}\gamma = \dfrac{\frac{V_f - V_i}{V_i}}{\Delta T} {/eq}.

For a body with an equal linear coefficient in all three directions( length, breadth, and height), the relation can be approximated as

{eq}\gamma = 3\alpha {/eq}.

The volumetric coefficient of thermal expansion of water at {eq}20^{\circ} C {/eq} is

{eq}\gamma = 3\alpha = 3* 69 * 10^{-6} K^{-1} = 207 * 10^{-6} K^{-1} {/eq}.

Now, we can write

{eq}207 * 10^{-6} = \dfrac{\frac{V_f - V_i}{V_i}}{\Delta T} = \dfrac{\frac{V_f - 1}{1}}{21-20} \,\,\Rightarrow\,\, V_f = 1+ 207 * 10^{-6} {/eq} m^3

The mass of water does not change with temperature. Hence, we can write

{eq}\rho_i V_i = \rho_f V_f {/eq} where {eq}\rho {/eq} is the density of water.

Initially, the density of water is {eq}1000 {/eq} kg/(cubic m).

We also know the initial and final volume of water. Hence,

{eq}1000(1) = \rho_f (1+ 207 * 10^{-6} ) \,\,\Rightarrow\,\, \rho_f = \dfrac{1000}{1+ 207 * 10^{-6}} = 999.793 {/eq}.

The percentage change in density is {eq}\dfrac{\rho_f - \rho_i}{\rho_i} * 100 \% = \dfrac{ 999.793- 1000}{1000} * 100 \% = -0.02 \% {/eq}.