Question

1 gram of steam is sent into one gram of ice. At thermal equilibrium the resultant temperature of the mixture is:
A.) $270^o$C
B.) $230^o$C
C.) $100^o$C
D.) $120^o$C

Answer 1

Hint: Check the amount of heat required for ice to melt and check the amount of heat required for steam to get condensed completely and then calculate the amount of heat required for the melted ice to get converted into the water at hundred degree Celsius and comparing the heat required to condense the steam determine the final temperature of the mixture.

Step-by-step solution:
The total heat gained by the ice should be equal to the total heat lost by steam and let us calculate the heat required for the ice to completely convert into water

The heat required for the ice to completely convert into water is given by mL where M is the mass of the ice is the latent heat

So heat required = 1 $\times$ 80 = 80 cal

The amount of the heat released when to convert the 1 g of steam to water at 100-degree celsius is obtained as mL where m is the mass of the steam and the L is the latent heat of the vaporization

So we get the heat released as mL = 1 $\times$ 540 = 540 cal

Now we can see that the amount of heat gained by the ice to convert into the water at zero degree celsius is not enough for the steam to get condensed completely.

Let us now calculate the amount of heat the melted ice or water at 0-degree celsius require to convert into the water at 100-degree Celsius :

We know for the heat transfer $m S \Delta T = \Delta Q$, where m is the mass of the water and S is the specific heat of the water and T is the temperature of the water.
So we get $ms\Delta T = 1\times 1 \times (100-0) = 100$ cal
So we can see the total heat required for the ice to melt and become water at 100-degree celsius becomes 100 + 80 = 180 calories

We can see that the amount of heat to be absorbed by ice is still larger as the heat released by the steam is 540 calories.

So the ice after becoming water at 100 degrees Celsius, some amount of water gets converted into steam and some amount of steam gets converted into water and the final mixture temperature becomes 100 degrees Celsius. The final mixture contains both water and steam at a hundred degrees Celsius.

So the final temperature of the mixture is obtained as $100^o C$

Note: In this kind of problem we cannot directly use the law of method of mixtures as the state conversions are involved. We need to check the amount of heat released and absorbed converting into a different state and then finally find the state of the mixture and the temperature of the mixture.