Question

How much heat energy must be added to a 6.0-cm-diameter copper sphere to raise its temperature from -50°C to 150°C?

Short answer

$Q=77.77[kJ]$

Step-by-step solution

Given information

The diameter of the sphere is $6.0\mathrm{cm}$and the temperature goes from $-50°\mathrm{C}$to $150°\mathrm{C}$.

Simplification

The heat energy can be calculated with the heat capacity equation:

$Q=m\Delta C_p\Delta \Delta T$

Where $m$is the mass, $C_p$is the specific heat and $\Delta T$is the difference of temperature in the object. The change of heat energy is positive when the temperature increase and negative if otherwise. To calculate the mass of the sphere:

$m=V\Delta \rho$

Where $V$is the volume and $\rho$is density.

Calculation:

First, we need to know how much mass of copper we have; from copper property tables we find its density:

$\rho =8960\left[\frac{\mathrm{kg}}{{\mathrm{m}}^3}\right]$

Substituting values:

$m=\frac{4}{3}\pi (0.03[\mathrm{m}]{)}^3\Delta 8960\left[\frac{\mathrm{kg}}{{\mathrm{m}}^3}\right]=1.01[\mathrm{kg}]$

The specific heat of copper:

$C_p=0.385\left[\frac{kJ}{kg{\Delta }^{D}C}\right]$

Solving for $Q$:

$Q=1.01[\mathrm{kg}]\Delta 0.385\left[\frac{\mathrm{kJ}}{k_g\Delta °\mathrm{C}}\right]\Delta \left(150\left[{}^{\circ }\mathrm{C}\right]-\left(-50\left[{}^{\circ }\mathrm{C}\right]\right)\right)=77.77\left[kJ\right]$