Answer to Question #160110 in General Chemistry for Madison

Question

A 100.0 g sample of water at 27.0°C is poured into a 77.400000000000006 g sample of water at 89.0°C. What will be the final temperature of the water?

Answer 1

Q160110

 

A 100.0 g sample of water at 27.0°C is poured into a 77.400000000000006 g sample of water at 89.0°C. What will be the final temperature of the water?

Solution:

We are given two samples of water. One is at room temperature (27.0°C) and the other is hot (89.0°C).

Sample 1: 100.0 g sample of water at 27.0°C ( Room temperature / cold water )

Sample 2: 77.4 g sample of water at 89.0°C. ( Hot water )

Even though the mass of hot water is given in 17 significant figures. There is no need to include all digits in the calculation. We can just use 77.4 g in the calculation.

When we mix the two samples of water, the water present at room temperature will gain heat and

hot water will lose heat.

The mixture will have a certain temperature in between 27.0°C and 89.0°C. The temperature of the mixture will depend on the mass of the cold and hot water. Let us consider that the final temperature of the mixture is 'x'

Heat gained by cold water = m_{cold water} * s *  ΔT _{cold water}

= 100.0g * 4.184 J/g °C * ( x - 27.0 °C)

Heat loss by hot water = m _{hot water} * s * ΔT _{hot water}

= 77.4 g * 4.184 J/g °C * ( x - 89.0 °C)

Heat gained by cold water = - Heat loss by hot water

( Hot water is losing heat so a negative sign is written here.)

100.0g * 4.184 J/g °C * ( x - 27.0 °C) = - 77.4 g * 4.184 J/g °C * ( x - 89.0 °C)

Cancel out 4.184 J/g °C from both the side.

100.0g * ( x - 27.0 °C) = - 77.4 g * ( x - 89.0 °C)

100.0g * ( x - 27.0 °C) = 77.4 g * ( 89.0 °C -x )

Let us ignore the units and do the calculation

100x - 2700 = 6888.6 - 77.4 x

100 x + 77.4 x = 6888.6 + 2700

177.4x = 9588.6;

divide both the side by 177.4, we have

177.4x/ 177.4 = 9588.6 / 177.4 ;

x = 54.051 °C

In 3 significant the answer will be 54.1 °C.

I hope that this will help you understand the problem.

Please let me know if you have not understood any step.

Thank you. :)