Math used in this equation rearrangement?

[MarchON]

I'm trying to determine how much water it takes to raise a car's temperature from -25°C to 0°C. The water is at 10°C.

What I apparently need to have set up is:

-ΔU_int,water = ΔU_int,car

-(m_wc_wΔT_w) - m_wL_f,w = m_cc_cΔT_c

The resultant rearranged equation looking for mass of water gives this:

m_w = -(m_cc_cΔT_c)/c_wΔT_w - m_wL_f,w

I don't understand how this was done. Also, why are you subtracting Latent heat of fusion x Mass from m_wc_wΔT_w?

 

[NascentOxygen]

Looks like there's a mistake in the algebra, then. What do you think that eqn should be?

Q for you: why does latent heat of fusion enter into the picture at all?

 

[jtbell]

The resultant rearranged equation looking for mass of water gives this:

mw = -(mcccΔTc)/cwΔTw - mwLf,w

Says who?

(Hint: are the units consistent?)

 

[MarchON]

Latent heat of fusion is in the picture because the water is going from a liquid to a solid. It freezes when it hits the car, then at a certain point it doesn't because the car warms up to 0 degrees. And I don't know what's up with the equation, but that's what my professor's solution says. Based on my math, I got something that makes no sense:

0= m_carc_carΔT_car/C_wΔT_w + L_f

 

[NascentOxygen]

The left side should be m_w. You need a pair of brackets on the right side, and then it should look right.

 

[MarchON]

I don't understand how. Is there any way (and I know this is no easy task) to break down the algebra step by step for me?
Also, I made a mistake with the resultant equation in my first post. It's actually mw = -(m_cc_cΔT_c)/c_wΔT_w - Lf,_w (no - m_wL_f,w)

I realize that ends up being the same thing that you said and there is no error (but he kept the negatives in, whereas we canceled them out), but I still don't understand how.

 

[NascentOxygen]

You started with this: -m_wc_wΔT_w - m_wL_f,w = m_cc_cΔT_c
Taking out a common factor -m_w we have
-m_w(c_wΔT_w +L_f,w)= m_cc_cΔT_c

Now divide both sides by (c_wΔT_w +L_f,w)
and we are left with
-m_w = m_cc_cΔT_c / (c_wΔT_w +L_f,w)

Multiplying both sides by -1 so that we end up with m_w by itself,
m_w = -m_cc_cΔT_c / (c_wΔT_w +L_f,w)

The brackets I said you needed are those in the denominator; the ones you added in the numerator make no difference.