A block of ice of mass 0.7 kg and initial temperature T = 0 o C is placed in a s

Question

A block of ice of mass 0.7 kg and initial temperature T = 0 oC is placed in a sealed insulated container full of Helium gas, initially at temperature 155 oC and pressure of 1 atm.   The volume of the Helium is 760 L, and is constant. Helium is a monatomic ideal gas.

What is the mass of the liquid water when the system comes to equilibrium? (In other words, how much ice melts?)

Assume no heat is lost to the surroundings.

Give your answer in kg to three significant digits. Do not include units in your answer.

Explanation / Answer

Let the equilibrium temperature be Teq.

In the condition given, heat will flow into ice system according to the equation:

Q1 = mLvap + mCwater delta T
Q1 = 0.7 x 334 + 0.7 x 4.186 x (Teq-0)
Q1 = 233.8 + 2.9302 Teq ---------------------- 1

This heat is heat ejected by the ideal gas Helium system, which in an isochoric process( constant volume, given condition) is given by the equation:

Q2 = nCv delta T
Q2 = n(3R/2) (155-Teq) ---------------------- 2

Now, using ideal gas equation to get the moles of helium in the container:

pV = nRT
n = pV/RT
n = 1 x 760 /(R x 428)
n = 190/107R

Plugging this in the equation 2:

Q2 = n(3R/2) (155-Teq)
Q2 = (190/107R)(3R/2)(155-Teq)
Q2 = 285 (155-Teq) / 107

Since Q1 = Q2 :

233.8 + 2.9302 Teq = 285 (155-Teq) / 107
25016.6 + 313.5314 Teq = 44175 - 285Teq
598.5314 Teq = 19158.4
Teq = 32 degree Celsius

Mass of liquid = 0.7 kg

Answer: 0.7

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