Calculating Time to Heat Earth to 100°C

[djk628]

Homework Statement

The Earth intercepts 1.27 x 10^17 W of radiant energy from the Sun. Suppose the Earth, of volume 1.08 x 10^21 m3, was composed of water. How long would it take for the Earth at 0°C to reach 100°C, if none of the energy was radiated or reflected back out into space?

Homework Equations

Q=mc*deltaT

The Attempt at a Solution

I just want to make sure that I did this correctly because the answer choices are all similar and vary by degrees of magnitude and I would like to avoid a simple mistake. Multiply the volume by 1000 to get the weight in kilograms, then multiply by 100, the temperature change. Multiply this by the specific heat. So:

1.27 x 10^17 = 1.08 x 10^24 kg * 4186 * 100

divide both sides by Q, gives a value in seconds. Convert to years which is approximately 113 years. Is this correct?

 

[rock.freak667]

your equation should be

(1.27 x 10^17)*t = 1.08 x 10^24 kg * 4186 * 100

But other than that, it looks like it should be correct.

 

[nlightner]

I would like to commend you for your attempt at solving this problem using the appropriate equation (Q=mc*deltaT) and for double-checking your work to avoid simple mistakes. Your solution is mostly correct, but there are a few things that can be improved upon.

Firstly, your calculation of the weight in kilograms is incorrect. The volume of the Earth is given in cubic meters, so it should be multiplied by the density of water (1000 kg/m^3) to get the weight in kilograms. Therefore, the correct weight would be 1.08 x 10^24 kg.

Secondly, the specific heat you used (4186) is the specific heat of liquid water. However, the Earth is not entirely composed of liquid water, so it would be more accurate to use the average specific heat of the Earth's crust, which is approximately 1000 J/kg*K. This would give a more realistic estimate of the time it would take for the Earth to heat up.

Finally, your final answer of 113 years is approximately correct, but it would be more accurate to convert the time from seconds to years with more precision. The exact answer is approximately 112.5 years.

In conclusion, your approach to solving this problem was correct, but there were a few minor errors that could be improved upon for a more accurate solution. Keep up the good work!