Chapter 14, Problem 12P

To determine

Ratios of masses of copper, aluminium and water.

Answer to Problem 12P

Solution:

Ratios of masses of copper, aluminium and water is $1:4.54:11.13$

Explanation of Solution

Given:

$cpw$ of water is 4186 J/kgK

$cpa$ of aluminium is 921 J/kgK

$cpc$ of copper is 376 J/kgK

$mw$ of water

$ma$ of aluminium

$mc$ of copper

Formula Used:

The formula used is the relation between temperature and specific heat from thermodynamics

$Q=mcp(Tf-Ti)$

Where,

$Q$ is the heat transferred

$cp$ is the specific heat

$m$ is the mass

$Tf$ is the final temperature

$Ti$ is the initial temperature

Calculation:

Since all the three materials rise to the same temperature for the same amount of heat, the quantities $Q$ and $(Tf-Ti)$ are the same.

One can use the heat capacity formula: $Q=mcp(Tf-Ti)$

for each material:

For water: $Q=mwcpw(Tf-Ti)$

For aluminium: $Q=macpa(Tf-Ti)$

For copper: $Q=mccpc(Tf-Ti)$

The ratio of water to aluminium is given by dividing the first two equations above.

$\frac{Q}{Q}=\frac{mwcpw(Tf-Ti)}{macpa(Tf-Ti)}$

$1=\frac{mwcpw}{macpa}$

$macpa=mwcpw$

$\frac{ma}{mw}=\frac{cpw}{cpa}$

Substituting the given values,

$\frac{ma}{mw}=\frac{4186}{921}=4.54$

In the same fashion one can find for water and copper.

$\frac{mc}{mw}=\frac{cpw}{cpc}$

Substituting the given values,

$\frac{mc}{mw}=\frac{4186}{376}=11.13$

If $mw$ is 1 then $ma$ is 4.54 and $mc$ is 11.13.