Some gun owners make their own bullets, which involves them melting lead and casting it into lead slugs. A 0.198-kilogram mass of lead at an initial temperature of 7.00 degrees Celsius is heated to its melting point temperature of 327 degrees Celsius and completely melted. Determine the heating required to do this. Use a value of 128 joules per kilogram degrees Celsius for the specific heat capacity of lead and use a value of 24.5 kilojoules per kilogram for the specific latent heat of fusion of lead.
We can call the amount of heating required to make this transformation capital 𝑄. In this example, we start out with a chunk of lead, whose mass we know, and we also know its initial temperature 7.00 degrees Celsius. We then start to heat this mass of lead, raising its temperature eventually to 327 degrees Celsius. Not only do we raise the temperature of the lead, but we also change its state from a solid to a completely melted liquid.
Finding out how much total heating this requires involves two steps. First, we’ll figure out how much heat we added in order to change the temperature of the lead from its initial to its final temperature. And then, we will figure out how much heating was needed to change the lead from a solid to a liquid.
We can say then that the total heating will equal the heat that’s needed to increase the temperature of the lead plus the heat that’s needed to melt it. Focusing on this first term, that has to do with the specific heat capacity we’ve called it 𝐻 sub 𝑐 of lead. This specific heat capacity tells us how much energy is needed in order to raise the temperature of one kilogram of lead one degrees Celsius.
To find out what this total heat is in joules, we’ll want to multiply this specific heat capacity by just how much lead we have in kilograms and the temperature increase in degrees Celsius. The chunk of lead we’re working with has a mass of 0.198 kilograms. And as far as temperature change, we know that’s equal to the final temperature minus the initial temperature of the lead.
Considering the units of this entire expression, see how the units of kilograms as well as the units of degrees Celsius cancel out, leaving us with a result in units of joules. When we calculate this heat needed to increase the temperature of the lead, we find that it’s about 8110 joules.
Interestingly, if we stopped added heat at this point, then we would have heated our solid mass of lead from 7.00 degrees Celsius to 327. But it would still be completely solid. We need to add heat energy on top of this in order to change the state or the phase of this lead from a solid to a liquid. That’s where the heat of fusion comes in.
The heat of fusion tells us just how much energy is needed for one kilogram of our particular material in order to change the phase of that material from a solid to a liquid. To find out the heating needed to make this change, we’ll take the heat of fusion of our material and we’ll multiply it by the mass of this chunk of lead.
When we do, notice that the units of kilograms cancel out in this expression. And we’re left with units of joules, heat energy. This product is approximately 4851 joules. And we now have an expression for the total heat energy needed to melt this lead.
To three significant figures, 𝑄 is 13.0 kilojoules. That’s the energy that will be needed to heat and then melt this chunk of lead.