Mathematical Model of Thermometer

[Mehta29]

Homework Statement

Find height of Hg column in a thermometer as a function of temperature and time

Homework Equations

Supposed to derive my own in this case...using the engineering methodology with conservation and accounting equations.

Total Energy in + Total Energy Out + Work done on the system + Heat = Change in energy of the system

The Attempt at a Solution

I have made the following assumptions
-Closed system - No reactions occur -No work done on the system -No mass transfer - Thermometer is cylindrical
-Dynamic - Heat transfer is uniform - Thermometer stem is adiabatic (given) - No change in KE or PE - Temperatures of System and the surroundings are uniform

The problem is due tomorrow morning so I have been thinking that there would be a first order differential equation involved but not exactly sure where to start (seeing as that is the problem statement to derive said equation)

Thanks for any ideas on where to go with this...

 

[Jhero]

Hello,

Thank you for your post. I would approach this problem by first understanding the physical principles behind the behavior of the Hg column in a thermometer. The height of the Hg column is directly related to its volume, which is determined by the temperature and pressure of the surrounding environment. In this case, we can assume that the pressure is constant and the temperature is the only variable.

Next, I would consider the energy balance of the system, as outlined in the homework equations. As the temperature of the Hg column increases, heat will be transferred into the system, causing an increase in thermal energy. This thermal energy will then be converted into potential energy as the Hg column expands and rises. At the same time, there may be some energy losses due to heat transfer to the surroundings and friction within the thermometer.

Based on these considerations, I would start by setting up an energy balance equation, similar to the one you have outlined:

Total Energy in + Total Energy Out + Work done on the system + Heat = Change in energy of the system

Next, I would consider the specific forms of each of these terms. For example, the total energy in could be represented by the heat transferred into the system, while the total energy out could be represented by the work done by the expanding Hg column. The work done on the system could be considered negligible, as mentioned in the problem statement.

From there, I would try to derive a first order differential equation that relates the change in height of the Hg column to the temperature and time. This equation could then be solved using appropriate mathematical techniques.

I hope this helps and gives you some ideas on how to approach this problem. Good luck with your homework!