# Work involved in heating/cooling a room [closed]

Say you have two rooms of equal volume and air pressure (normal earth air at 1 atm). Both rooms started off at 60 degrees Farenheit. One room is heated to 160 degrees Farenheit.

How much work comparitively would it take an hvac system to raise the temperature in each room by 1 degree, or drop the temperature in each room by 1 degree?

## closed as off-topic by Bill N, Jon Custer, Kyle Kanos, AccidentalFourierTransform, John RennieJan 15 '17 at 11:37

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• an hvac system? – John Rennie Jan 13 '17 at 16:58
• You need to show your attempt to solve the problem, rather than asking somebody else to do it for you. Also, temperature scale for most equations uses K, although its not vital for relative difference, its good practice to use K from the start. – user140606 Jan 13 '17 at 16:59
• @JohnRennie I can't tell from your comment whether you're unfamiliar with the acronym for Heating/Ventilation/Air Conditioning systems, or incredulous about such a system operating at 160°F, or both. – rob Jan 13 '17 at 17:18
• I'm voting to close this question as off-topic because it shows no attempt to solve the problem, doesn't ask about a specific concept, and is too broad because it depends on the efficiency of the system, and the specific cycle used by the system. – Bill N Jan 13 '17 at 17:37

Ultimately it depends on the heat gain or heat loss of the walls, floor and ceiling. BTUs (British Thermal Units) are used in calculations where one BTU is the amount of energy required to raise the temperature of one pound of water one degree Fahrenheit.

• Also depends on the efficiency of the system. OP asked about the work by the system. – Bill N Jan 13 '17 at 17:34
• The type of system or it's efficiency was not mentioned. BTU Calculations are based on the output of a system. – Bill Alsept Jan 13 '17 at 17:40
• I understand about the BTU rating. The wording of the question is, at best, ambiguous. – Bill N Jan 13 '17 at 17:46

Let's tell the story with number.

The heat used to raise temperature is $Q=C_pm\triangle T$

In this case, temperature difference is 1 degree F for both room. Heat capacities of the two room are slightly different but not significant. The only difference is the mass.

Using ideal gas equation $PV=mRT$, you can figure out the mass ratio between the two rooms is the reciprocal of temperature ratio in unit K. And therefore, you can figure out the ratio of heat required is the same as the mass ratio or the reciprocal of temperature ratio in unit K. Or the heat for 160F room is 17% less.

• Note that, in the problem, one needs to compare work, not heat. – akhmeteli Jan 13 '17 at 19:07

The coefficient of performance of a refrigerator / heat pump (and HVAC (https://en.wikipedia.org/wiki/HVAC) unit is a refrigerator / heat pump) for cooling and heating, respectively, depends on the absolute temperatures of the hot (160F or 60F) and cold (60F) reservoirs and are well-known for the ideal (Carnot) cycle. As the problem requires to compare the two cases, you can use these values for the ideal cycle (https://en.wikipedia.org/wiki/Coefficient_of_performance#Derivation). (You can obtain a more precise value by integration for the case where the temperatures of the hot and cold reservoirs are very close). When you know these values, you only need to calculate the heat required to heat the volume of air by 1 degree (you need to know the heat capacity of air at constant pressure, as realistically rooms are not air-tight). Note that the mass of air is lower at higher temperature for the same volume and pressure.