The Science Snail
  • Home
  • Science
  • Philosophy
  • About
  • Contact
  • Privacy Policy

The molecular difference between heat and work

8/14/2013

2 Comments

 
Heat and work represent two distinct means by which the internal energy of a system can change. In this article I clarify the fundamental, molecular differences between these two processes. Examples of both heat and work are abundant in everyday life. ice melts at room temperature because heat flows from surrounding air to the ice. Similarly, work must be done on gas in a piston to compress it. Both of these processes involve increasing the energy of the system. In fact, the first law of thermodynamics states that the change in internal energy of a system is equal to the amount of heat and work done. Interestingly, internal energy is a state function (it depends only on the present state of the system), while the heat and work are path functions (they depend on the process through which the change occurred). Thus for a reversible process, the first law is given by
Picture
Where U is the internal energy, w is the reversible work, and q is the reversible heat. A very common form of work is pressure-volume work, such as compression. In this case the work can be expressed as
Picture
Where P is the pressure and V is the volume of the system. Although it is easy to distinguish between examples of work or heat, it is less obvious how these quantities are fundamentally different. Consideration of the system on a molecular level will help with this. From this prospective, there are a variety of discrete energy states that the system can occupy. The macroscopically observed energy will be equal to the average energy of the system. From probability, the average value of a discrete random variable can be expressed as the sum of the product of each value and its respective probability. We can apply this to the internal energy of a macroscopic system since it’s allowed energy states are discrete.
Picture
Where U is the macroscopically observed internal energy, p is the probability of the system being in state j and E is the state's energy. Do not worry about how the probability is determined yet, I discuss this in my article about temperatures below absolute zero.

If the above equation is differentiated using the product rule, it becomes
Picture
E is a function of the number of particles (N) and the volume (V). Since the number of particles is constant, dE can be viewed as the change in energy due to a small change in volume. This implies that
Picture
This expression is analogous to the first law of thermodynamics! I restate this law below: 
Picture
 The work (-PdV) must correspond to the first term above, because both are in terms of dV. Since there are only two terms, the second term must give the heat. Thus:
Picture
This result finally shows how heat and work are different on the molecular level. Work changes the amount of energy that each allowed state has, without altering the probability distribution of the states (p is constant). Conversely, heat does not change the allowed energy states of the system (E is constant). Instead, it changes the probability distribution of the allowed states. They both change the average energy, but do so in different ways.

2 Comments
Clinton Morrison link
8/14/2013 06:08:52 am

Interesting! I never understood what the difference was until now!

Reply
ALAN EVANSON
10/19/2020 12:25:34 am

This is one of the most informative answers that I have ever read. How does the above analysis help explain electrical work? My understanding is that if a current runs through a resistor, electrical work is being done on the resistor, even though it will become hotter as if heat was applied. How can we use the above to explain that it is work and not heat that is being applied to the resistor?

Reply



Leave a Reply.

    Archives

    December 2022
    April 2022
    December 2021
    December 2020
    September 2020
    August 2020
    December 2019
    June 2019
    February 2019
    January 2019
    December 2018
    October 2018
    August 2018
    July 2018
    February 2018
    December 2017
    December 2016
    April 2016
    January 2016
    August 2015
    July 2015
    February 2015
    December 2014
    August 2013

    Categories

    All
    Biochemistry
    Biology
    Enzymology
    Finance
    Health
    Mathematics
    Organic Chemistry
    Physical Chemistry

    RSS Feed

    List of all articles

    Mathematics of compound interest

    Dynamic nuclear polarization in solid state NMR 

    Modelling a multi-state G protein signalling pathway

    Organic synthesis of Aspirin

    Distinguishing enzyme inhibition mechanisms


    ​Metformin total synthesis
    ​
    ​Ki, Kd, IC50, and EC50 values

    ​
    AZT: mechanism and synthesis


    Forecasting website ad revenue

    Km vs Kd

    The relationship between TV screen size and price

    ​Health benefits of green tea

    Synthesis of ibuprofen from benzene

    ​The mechanism of action of Eflornithine
    ​
    Synthesis of sucralose from sucrose


    Diluting a solution to Avogadro's limit
    ​
    The affect of mutation rate on evolutionary equilibrium

    ​
    Organic synthesis of indomethacin


    Probing protein-protein interactions in the yeast glycolytic metabolon

    Time course enzyme kinetics

    A generalized​ model for enzymatic substrate inhibition
    ​
    ​The basis of high thermostability in thermophilic proteins
    ​

    First order drug elimination kinetics
    ​
    Improving the efficiency of protein dialysis: constant dialysate replacement


    Mathematical modelling of evolution

    Calculating the optimum ddNTP:dNTP ratio in Sanger sequencing

    A mathematical model of hair growth

    Life does not violate the second law of thermodynamics


    Collagen and the importance of vitamin C

    Temperatures below absolute zero are surprisingly hot

    The molecular difference between heat and work

    Want to keep up to date with new articles? Subscribe to the monthly newsletter! ​

Subscribe to Newsletter

​© 2020 Copyright The Science Snail. All rights reserved.