A copper cup of 100 cm^{3} capacity is fully filled with glycerin at 25^{\circ}C . What is the...

Question:

A copper cup of 100 {eq}cm^{3} {/eq} capacity is fully filled with glycerin at {eq}25^{\circ}C {/eq}. What is the temperature required to spill out the 2.52 {eq}cm^{3} {/eq} of glycerin from cup. The coeffecient of volume expansion of glycerin is {eq}5.1\times 10^{-4} {/eq}/ celsius degree and the coeffecient of linear expansion of copper is {eq}17.6\times 10^{-4} {/eq} / celsius degree.

Volume Expansion:

Volume expansion is the property of the metal, which defines that the rise of temperature metal will expand volumetrically. Every metal has a different value of the coefficient of volume expansion. Considering this property, we use the metal in the design field.

Given data

• The volume of cup is: {eq}V = 100\;{\rm{c}}{{\rm{m}}^{\rm{3}}} {/eq}
• The initial temperature is: {eq}T = 25^\circ \;{\rm{C}} = 298\;{\rm{K}} {/eq}
• The spill out volume is: {eq}\Delta V = 2.52\;{\rm{c}}{{\rm{m}}^{\rm{3}}} {/eq}
• The coefficient of volume expansion of glycerin is: {eq}5.1 \times {10^{ - 4}}\;{{\rm{C}}^{{\rm{ - 1}}}} {/eq}
• The coefficient of linear expansion is {eq}17.6 \times {10^{ - 4}}\;{{\rm{C}}^{{\rm{ - 1}}}} {/eq}

Expression to find the spill out volume,

{eq}\Delta V = \left( {{\beta _G} - {\beta _A}} \right)V\Delta T {/eq}

Here, {eq}\Delta V {/eq} is the spill out volume, {eq}{\beta _G} {/eq} is the coefficient of volume expansion of glycerin and {eq}{\beta _A} {/eq} is the coefficient of volume expansion of the aluminum.

Substituting all the value in the above equation,

{eq}\begin{align*} 2.52 &= \left( {5.1 \times {{10}^{ - 4}} - 17.6 \times {{10}^{ - 4}}} \right) \times 100 \times \left( {25 - {T_f}} \right)\\ 25 - {T_f} &= \dfrac{{2.52}}{{\left( {5.1 \times {{10}^{ - 4}} - 17.6 \times {{10}^{ - 4}}} \right) \times 100}}\\ 25 - \dfrac{{2.52}}{{\left( { - 0.125} \right)}} &= {T_f}\\ {T_f} &= 45.16^\circ \;{\rm{C}} \end{align*} {/eq}

Thus, the final required temperature is {eq}45.16^\circ \;{\rm{C}} {/eq}