First Law of Thermodynamics Summary

• Internal Energy

  Internal energy is all the mechanical energy in all the components of a system. For example, in a monoatomic gas it might be the sum of the kinetic energies of all the gas atoms. In a solid it might be the sum of the kinetic and potential energies of all the particles that make up the solid.

• Heat

  Heat is a bit more complicated. It is internal energy as well, but it is internal energy that is transferred into or out of a given system. Furthermore, it is in some fundamental sense ``disorganized'' internal energy - energy with no particular organization, random energy. Heat flows into or out of a system in response to a temperature difference, always flowing from hotter temperature regions (cooling them) to cooler ones (warming them).

  Common units of heat include the ever-popular Joule and the calorie (the heat required to raise the temperature of 1 gram of water at 14.5$^\circ$ C to 15.5$^\circ$ C. Note that 1 cal = 4.186 J. Less common and more esoteric ones like the British Thermal Unit (BTU) and erg will be mostly ignored in this course; BTUs raise the temperature of one pound of water by one degree Fahrenheit, for example. Ugly.

• Heat Capacity

  If one adds heat to an object, its temperature usually increases (exceptions include at a state boundary, for example when a liquid boils). In many cases the temperature change is linear in the amount of heat added. We define the heat capacity $C$ of an object from the relation:
  

  $$\Delta Q = C \Delta T$$ (219)

  
  
  where $\Delta Q$ is the heat that flows into a system to increase its temperature by $\Delta T$. Many substances have a known heat capacity per unit mass. This permits us to also write:
  

  $$\Delta Q = m c \Delta T$$ (220)

  
  
  where $c$ is the specific heat of a substance. The specific heat of liquid water is approximately:
  

  $$c_{\rm water} = 1 {calorie/gram-^\circ C}$$ (221)

  
  
  (as one might guess from the definition of the calorie above).

• Latent Heat As noted above, there are particular times when one can add heat to a system and not change its temperature. One such time is when the system is changing state from/to solid to/from liquid, or from/to liquid to/from gas. At those times, one adds (or removes) heat when the system is at fixed temperature until the state change is complete. The specific heat may well change across phase boundaries. There are two trivial equations to learn:
  

  $$\Delta Q_f = m L_f$$ (222)

  $$\Delta Q_v = m L_v$$ (223)

  
  
  where $L_f$ is the latent heat of fusion and $L_v$ is the latent heat of vaporization. Two important numbers to keep in mind are $L_f(H_2 O) = 333$ kJ/kg, and $L_v(H_2 O) = 2260$ kJ/kg. Note the high value of the latter - the reason that ``steam burns worse than water''.

• Work Done by a Gas
  

  $$W = \int_{V_i}^{V_f} PdV$$ (224)

  
  
  This is the area under the $P(V)$ curve, suggesting that we draw lots of state diagrams on a $P$ and $V$ coordinate system. Both heat transfer and word depend on the path a gas takes $P(V)$ moving from one pressure and volume to another.

• The First Law of Thermodynamics
  

  $$\Delta E_{\rm int} = \Delta Q - W$$ (225)

  
  
  In words, this is that the change in total mechanical energy of a system is equal to heat put into the system plus the work done on the system (which is minus the work done by the system, hence the minus above).

  This is just, at long last, the fully generalized law of conservation of energy. All the cases where mechanical energy was not conserved in previous chapters because of nonconservative forces, the missing energy appeared as heat, energy that naturally flows from hotter systems to cooler ones.

• Cyclic Processes Most of what we study in these final sections will lead us to an understanding of simple heat engines based on gas expanding in a cylinder and doing work against a piston. In order to build a true engine, the engine has to go around in a repetitive cycle. This cycle typically is represented by a closed loop on a state e.g. $P(V)$ curve. A direct consequence of the 1st law is that the net work done by the system per cycle is the area inside the loop of the $P(V)$ diagram. Since the internal energy is the same at the beginning and the end of the cycle, it also tells us that:
  

  $$\Delta Q_{\rm cycle} = W_{\rm cycle}$$ (226)

  
  
  the heat that flows into the system per cycle must exactly equal the work done by the system per cycle.

• Adiabatic Processes are processes ($PV$ curves) such that no heat enters or leaves an (insulated) system.

• Isothermal Processes are processes where the temperature $T$ of the system remains constant.

• Isobaric Processes are processes that occur at constant pressure.

• Isovolumetric Processses are processes that occur at constant volume.

• Work done by an Ideal Gas: Recall,
  

  $$PV = NkT$$ (227)

  
  
  where $N$ is the number of gas atoms or molecules. Isothermal work at (fixed) temperature $T_0$ is thus:
  

  $$W = \int_{V_1}^{V_2} \frac{NkT_0}{V} dV$$ (228)

  $$= NkT \ln(\frac{V_2}{V_1})$$ (229)

  
  

  Isobaric work is trivial. $P = P_0$ is a constant, so
  

  $$W = \int_{V_1}^{V_2} P_0 dV = P_0 (V_2 - V_1)$$ (230)

  
  

  Adiabatic work is a bit tricky and depends on some of the internal properties of the gas (for example, whether it is mono- or diatomic). We'll examine this in the next section.