What Happens When a Hot Steel Ball is Dropped into Water or Ice?

[navm1]

Homework Statement

If a steel ball of mass 100g and 100°c is dropped into 1litre of water at 20°c, what is the temperature rise? What would happen if it were dropped into a mixture of ice and water at 0°c?

Homework Equations

Q=mcΔT
Q=ml_m where m_l is latent heat of melting
Q lost = Q gained

The Attempt at a Solution

Q_steel = (0.1) (420) (-80) = -3360J
the steel ball loses 3360J so the water must have gained 3360J

Q_water = 3360 = (1)(4200)ΔT

ΔT = 3360/4200=0.8°C
so the waters temperature has increased by 0.8 degrees.

For the second part it didnt specify how much ice so i just chose to have 1 litre of water with 150g of ice at 0°c.

Q_steel = (0.1)(420)(-100) = -4200J
so the steel ball has lost 4200J of heat and the water ice mixture will gain 4200J of heat

Q_water = Q_ice = ml_m

m_ice= Q_water/l_m = 4200/3.35x10⁵ = 0.0125kg

So only 12.5g of ice would be melted and the temperature wouldn't rise at all as the energy has gone into melting the ice and there is still a significant amount left.Am I on the right track here? Thanks

 

[Chestermiller]

The first part was not done correctly. The steel ball will not cool all the way down to 20 C. Both the steel ball and the water will end up at the same temperature.

Chet

 

[navm1]

i thought i had the right idea but the numbers came out unusual

(0.1)(420)(x-100) = (1)(4200)(x-20)

42(x-100) = 4200(x-20)
42x-4200 = 4200x -8400

then from here i got

(42x-4200x)-4200= -8400

-4158-4200= -8400

-4158x = 4200 - 8400

in the end i ended up with 100/99 which can't be right.

if i calculate it with

(0.1)(420)(x-100) = (1)(4200)(20-x)

42x-4200 = 84000-4200x
42x+4200x = 84000+4200
4242x=88200
88200/4242 = 20.79

instead, i get 20.79°c which makes more sense but i thought delta T was always final minus initial

edit: this also means that the steel ball loses 3326.4J instead

 

[Chestermiller]

The following equation is incorrect: (0.1)(420)(x-100) = (1)(4200)(x-20)

It should read: (0.1)(420)(100-x) = (1)(4200)(x-20)

The heat lost by the ball is gained by the water. Also, if you cancel before you multiply, the algebra is much simpler:

(100-x)=100 (x-20)

Chet

 

[navm1]

so now i have

(0.1)(420)(100-x) = (1)(4200)(x-20)

4200-42x = 4200x - 84000

4200 + 84000 = 4242x

88200/4242 = 20.79

(0.1)(420)(79.2) = 3326.4J

so same answer before but a positive 3326.4J as the answer making the temperature rise 0.8c . i was expecting it to be negative because the ball is losing energy.

thanks

 

[Chestermiller]

so now i have

(0.1)(420)(100-x) = (1)(4200)(x-20)

4200-42x = 4200x - 84000

4200 + 84000 = 4242x

88200/4242 = 20.79

(0.1)(420)(79.2) = 3326.4J

so same answer before but a positive 3326.4J as the answer. i was expecting it to be negative because the ball is losing energy.

thanks

The solution I get to your original equation is 19.2 C. You must have made a compensating mistake in the algebra somewhere.

Also, the 3326 J is the amount of heat lost by the ball.

 

[navm1]

I put the whole equation into wolfram alpha to double check and got the same answer too http://imgur.com/iatFmIJ

For the second question if I had 500g water and 500g ice would I use 0.5 for my mass of water or does the ice make it 1kg?

 

[Chestermiller]

I put the whole equation into wolfram alpha to double check and got the same answer too http://imgur.com/iatFmIJ

With all due respect to wolfram alpha, it can't be the same answer if you replace (x-100) with (100 - x).

For the second question if I had 500g water and 500g ice would I use 0.5 for my mass of water or does the ice make it 1kg?

Neither. After the steel ball is dropped in, a small amount of ice would melt to form liquid water, but the temperature of the ice water mixture would not change. All the heat from the steel ball would go into melting the small amount of ice.

chet

 

[navm1]

With all due respect to wolfram alpha, it can't be the same answer if you replace (x-100) with (100 - x).

Neither. After the steel ball is dropped in, a small amount of ice would melt to form liquid water, but the temperature of the ice water mixture would not change. All the heat from the steel ball would go into melting the small amount of ice.

chet

You said to use
(0.1)(420)(100-x) = (1)(4200)(x-20), if I use x-100 I get 19.2. Apologies for missing the point