$2\, kg$ of ice at $-20^{\circ}C$ is mixed with $5\, kg$ of water at $20^{\circ}C$ in an insulating vessel having a negligible heat capacity. Calculate the final mass of water remaining in the container. It is given that the specific heats of water and ice are $1\, kcal/kg/^{\circ}C$ and $0.5\, kcal/kg/^{\circ}C$ respectively,while the latent heat of fusion of ice is $80\, kcal/kg$.

$A$ thermos flask contains $250 \ g$ of coffee at $90^{\circ} C$. To this $20 \ g$ of milk at $5^{\circ} C$ is added. After equilibrium is established,the temperature of the liquid is (Assume no heat loss to the thermos bottle. Take specific heat of coffee and milk as $1.00 \ cal/g^{\circ} C$) (in $^{\circ} C$)

$A$ calorimeter contains $0.2 \ kg$ of water at $30^{\circ}C$. When $0.1 \ kg$ of water at $60^{\circ}C$ is added to it,the final temperature of the mixture becomes $35^{\circ}C$. What is the heat capacity of the calorimeter in $J/K$?

Steam at $100^{\circ}C$ is added slowly to $1400 \, \text{g}$ of water at $16^{\circ}C$ until the temperature of the water is raised to $80^{\circ}C$. The mass of steam required to do this is $(L_V = 540 \, \text{cal/g})$ ........... $\text{g}$

Steam at $100^{\circ}C$ is passed into $1.1 \, kg$ of water contained in a calorimeter of water equivalent $0.02 \, kg$ at $15^{\circ}C$ until the temperature of the calorimeter and its contents rises to $80^{\circ}C$. The mass of the steam condensed in $kg$ is:

Ice in a freezer is at $-7^{\circ} C$. $100 \, g$ of this ice is mixed with $200 \, g$ of water at $15^{\circ} C$. Take the freezing temperature of water to be $0^{\circ} C$,the specific heat of ice equal to $2.2 \, J/g^{\circ} C$,specific heat of water equal to $4.2 \, J/g^{\circ} C$,and the latent heat of ice equal to $335 \, J/g$. Assuming no loss of heat to the environment,the mass of ice in the final mixture is closest to .......... $g$.