Heat and Thermodynamics

When water (at temperature θ ) is mixed with ice (at 0∘ ), heat will flow from the warmer water to the ice.
The water cools down from θ to 0∘C.
The ice melts, using that heat.
No heat is lost to surroundings.
Step 1: Heat lost by water cooling to 0∘C
Q‌lost ‌=msθ
Step 2: Heat gained by ice (for melting)
If x is the mass of ice melted, then:
Q‌gained ‌=xL
Step 3: Equating heat lost and gained
msθ=xL
Step 4: Solve for x
x=‌

msθ

L

Step 5: Check if M (the large mass of ice) matters
Since M is very large, its temperature doesn't rise; it only partly melts.
Hence M appears only to indicate that the ice remains at 0∘C.
Final Answer:
x=‌

msθ

L