Question #8f7d4

1 Answer
Nov 5, 2017

47^@"C"

Explanation:

QUICK ANSWER

If you take T_f ""^@"C" to be the final temperature of the sample, you can say that you have

135 color(red)(cancel(color(black)("g"))) * (T_f - 21)color(red)(cancel(color(black)(""^@"C"))) = 82.5 color(red)(cancel(color(black)("g"))) * (89 - T_f)color(red)(cancel(color(black)(""^@"C")))

This will get you

T_f = (82.5 * 89 + 135 * 21)/(135 + 82.5)

color(darkgreen)(ul(color(black)(T_f = 47))) " " -> rounded to two sig figs

You can thus say that the final temperature of the sample will be 47^@"C".

color(white)(a)

THE DETAILED VERSION

Assuming that no heat is lost to the surroundings, you can say that the heat lost by the warmer sample will be equal to the heat gained by the colder sample.

color(blue)(ul(color(black)(q_"gained" = - q_"lost")))" "color(darkorange)("(*)")

The minus sign is used here because heat lost carries a negative sign.

Now, the amount of heat gained/lost depends on the mass of the substance, m, on its specific heat, c, and on the change in temperature, DeltaT, which is calculated by subtracting the initial temperature from the final temperature.

color(blue)(ul(color(black)(q = m * c * DeltaT)))

If you take T_f ""^@"C" to be the final temperature of the sample, you can say that the heat gained by the colder sample will be

q_"gained" = "135 g" * c_"water" * (T_f - 21)^@"C"

Similarly, the heat lost by the warmer sample will be

q_"lost" = "82.5 g" * c_"water" * (T_f - 89)^@"C"

Use equation color(darkorange)("(*)") to say that--do not forget about the minus sign!

135 color(red)(cancel(color(black)("g"))) * color(red)(cancel(color(black)(c_"water"))) * (T_f - 21) color(red)(cancel(color(black)(""^@"C"))) = - 82.5 color(red)(cancel(color(black)("g"))) * color(red)(cancel(color(black)(c_"water"))) * (T_f - 89)color(red)(cancel(color(black)(""^@"C")))

This simplifies to

135 * (T_f - 21) = - 82.5 * (T_f - 89)

which is equivalent to

135 * (T_f - 21) = 82.5 * (89 - T_f)

Once again, you will find that the final temperature of the sample will be 47^@"C".

The answer must be rounded to two sig figs, the number of sig figs you have for the two temperatures.