Mass of ice to warm a liquid - do I have it set up correctly?

[Linus Pauling]

1. An insulated beaker with negligible mass contains liquid water with a mass of 0.250 kg and a temperature of 76.1C.

How much ice at a temperature of -12.5C must be dropped into the water so that the final temperature of the system will be 33.0C? Take the specific heat of liquid water to be 4190 J/kg*K, the specific heat of ice to be 2100 J/kg*K, and the heat of fusion for water to be 334 kJ/kg.




2. Q = Mc deltaT
Q = +/- ML_f



3. warming the ice = Q1 = m_ice * c_ice * deltaT = m_ice * (2100) * (0 - 260.5K)

melting ice = Q2 = m_ice * L_f = m_ice * (334000 J/kg)

bring liquids to same temp. = Q3 = m_ice * c_water * deltaT = m_ice * 4190 * (306K - 0K)

cool water = Q4 = m_water * c_water * deltaT = 0.250 * 4190 * (306K - 349.1K)

Then, I did Q4 = Q1 + Q2 + Q3, and solved for m_ice:

-45147.25 = m_ice * (1282140 + 334000 - 547050)

m_ice = -0.0422, which I just took to be positive. I think I am at least on the right track, but have I set it up wrong since I get a negative number for mass?

 

[Linus Pauling]

Actually, I have to correct myself: I set the Q of cooling the liquid to final temperature equal to the other three (warming ice + melting ice + warming the new mass of liquid). Solving, I obtain the mass of the original ice to be 0.0906 kg.

Got it! Nevermind.

 

[kerimek]

I would first like to commend you for setting up the problem correctly and using the appropriate equations and values for specific heat and heat of fusion. However, I do see a mistake in your calculations that is leading to the negative mass value.

Firstly, when calculating the heat required to warm the ice, you have used a temperature difference of 0-260.5K, which is incorrect. The correct temperature difference should be 33.0C - (-12.5C), which is 45.5K. This will change the value of Q1 to be -94500 J.

Secondly, when calculating the heat required to bring the liquids to the same temperature, you have used a temperature difference of 306K - 0K, which is also incorrect. The correct temperature difference should be 306K - 33.0C, which is 273K. This will change the value of Q3 to be 0.250 * 4190 * 273 = 283747.5 J.

Using these corrected values in your equation Q4 = Q1 + Q2 + Q3, you will get a positive value for m_ice, which is the correct result.

Therefore, the correct mass of ice that must be added to the system is approximately 0.063 kg. I hope this helps clarify your calculations. Keep up the good work!