# With a fixed amount of water does it take less energy to raise the temperature for example from 10 degrees to 15 than from 40 to 45 degrees and if so, why?

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It takes slightly more heat energy to raise the temperature of a fixed amount of water from 10 to 15 degrees Celsius than from 40-45 degrees Celsius, at constant pressure, although the difference isn't likely to be measurable in the typical experiment.

The specific heat capacity of water, Cp, is usually considered a constant and is equal to 4.180 Joules/gram-degree C. This is the amount of heat needed to raise the temperature of one gram of water by one Celsius degree at 25 degrees C. However, it isn't really constant. Its temperature dependance is small over the range of temperatures that water can have as a liquid so it's generally neglected. Here's the approximate effect temperature has on Cp in these two ranges:

T = 10 degrees C, Cp = 4.192 J/g-deg C

T = 15 degrees C, Cp = 4.188 J/g-deg C

T = 40 degrees C, Cp = 4.179 J/g-deg C

T = 45 degrees C, C = 4.181 J/g-deg C

The specific heat capacity of water decreases from 0 degrees C to 35, then begins to increase again. The average of the Cp's for each pair of beginning and ending temperatures are:

10 and 15: 4.189 J/g-deg

40 and 45: 4.180 J/g-deg

Since the average is slightly higher for the lower temperature range, it will take more heat to raise the temperature in the lower range. The difference is most likely less that the degree of uncertainty in the thermometer used. It would also likely be very small compared to the additional heat loss that would occur at the higher temperature, even with a well-insulated system.

So in theory more heat would be required at the lower temperature but in practicality there would be no difference.