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Topic 19 of 19

Temperature, Heat & Thermodynamics

Thermodynamics describes energy at the macroscopic level — temperature, heat, and work. It governs everything from the efficiency of engines to the flow of heat through a building. The fundamental laws place absolute limits on what any heat engine can achieve and define the direction of natural processes.

Key Concepts

Temperature Scales
Celsius (°°C) and Kelvin (K) are related by TK=TC+273.15T_K = T_C + 273.15. Kelvin is the SI unit; absolute zero (0 K) is the lowest possible temperature, where molecular motion is minimal.
Thermal Expansion
Most materials expand when heated. Linear expansion: ΔL=αL0ΔT\Delta L = \alpha L_0 \Delta T (α = coefficient of linear expansion). Volume expansion: ΔV=βV0ΔT\Delta V = \beta V_0 \Delta T where β≈3α\beta \approx 3\alpha.
Ideal Gas Law
PV=nRTPV = nRT relates pressure PP, volume VV, amount nn (moles), and temperature TT (in kelvin). R=8.314R = 8.314 J/(mol·K). The ideal gas is a good model for real gases at low pressure and high temperature.
Heat and Specific Heat
Heat QQ is energy transferred due to a temperature difference. Q=mcΔTQ = mc\Delta T, where cc is the specific heat capacity (J/kg·K). For phase changes: Q=mLQ = mL (L = latent heat, no temperature change).
First Law of Thermodynamics
Energy conservation: ΔU=Q−W\Delta U = Q - W. Internal energy UU increases when heat flows in (Q>0Q > 0) and decreases when the system does work (W>0W > 0). This is the most general statement of energy conservation.
Second Law of Thermodynamics
Heat flows spontaneously only from hot to cold. Entropy (disorder) of an isolated system never decreases. No heat engine operating between temperatures THT_H and TCT_C can exceed the Carnot efficiency ηCarnot=1−TC/TH\eta_\text{Carnot} = 1 - T_C/T_H.

Key Equations

Ideal gas law
PV=nRT,R=8.314 J/(mol⋅K)PV = nRT, \quad R = 8.314 \text{ J/(mol·K)}
T must be in kelvin. n is moles. Also written PV = NkT where N is molecule count and k = 1.38×10⁻²³ J/K.
Heat and specific heat
Q=mc ΔTQ = mc\,\Delta T
c is the specific heat capacity of the material. For phase changes: Q = mL (no ΔT).
Thermal expansion
ΔL=αL0 ΔT,ΔV=βV0 ΔT\Delta L = \alpha L_0\,\Delta T, \quad \Delta V = \beta V_0\,\Delta T
α ≈ 10⁻⁵ to 10⁻⁶ K⁻¹ for most solids; β ≈ 3α.
First law of thermodynamics
ΔU=Q−W\Delta U = Q - W
ΔU = change in internal energy; Q = heat added to system; W = work done by system.
Carnot (maximum) efficiency
ηCarnot=1−TCTH\eta_\text{Carnot} = 1 - \frac{T_C}{T_H}
T in kelvin. No real engine can exceed this; it requires reversible processes.
Worked Example

Heating Water — Heat Calculation

Problem

How much heat is required to raise 2 kg of water from 20°C to 100°C? (cwater=4186c_\text{water} = 4186 J/kg·K.)

Solution
Q=mc ΔT=2×4186×(100−20)Q = mc\,\Delta T = 2\times4186\times(100-20)
Q=2×4186×80=669,760 J≈670 kJQ = 2\times4186\times80 = 669{,}760 \text{ J} \approx 670 \text{ kJ}
Answer ≈ 670 kJ of heat required.
Practice

Exercises

7 problems
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Exercise 1 / 7 Free
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An ideal gas at P₁ = 100 kPa, V₁ = 4 L is compressed isothermally to V₂ = 2 L. Watch the pressure bar change. Find P₂ using Boyle's law: P₁V₁ = P₂V₂.

P₂ (at V=2L) = kPa
Exercise 2 / 7 Free
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The P-V diagram shows an isobaric expansion at P = 200 kPa from V₁ = 2 L to V₂ = 4 L. Drag the handle to see the shaded work area. Find W = PΔV.

W = J
3 of 7

An ideal gas in a piston has n=2n = 2 mol, T=300T = 300 K, and pressure P=1.5×105P = 1.5\times10^5 Pa. What is the volume (in L)? Use R=8.314R = 8.314 J/(mol·K). (1 m³ = 1000 L.)

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4 of 7

A gas absorbs Q=500Q = 500 J of heat and does W=200W = 200 J of work on its surroundings. What is the change in internal energy (in J)?

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5 of 7

An ideal gas is compressed isothermally at T=350T = 350 K. The pressure doubles. By what factor does the volume change? (Enter the decimal factor, e.g. 0.5 for halved.)

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6 of 7

A Carnot engine operates between TH=600T_H = 600 K and TC=300T_C = 300 K. What is its maximum efficiency (as a percentage)?

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7 of 7

A heat engine operating between TH=800T_H = 800 K and TC=400T_C = 400 K absorbs QH=1200Q_H = 1200 J per cycle. How much work (in J) does a Carnot engine perform per cycle?

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Key Takeaways

  • Temperature in kelvin (TK=TC+273.15T_K = T_C + 273.15) is required for gas law and thermodynamics equations.
  • Ideal gas law PV=nRTPV = nRT: if any three variables are known, the fourth is determined.
  • Heat flows from high to low temperature; Q=mcΔTQ = mc\Delta T for sensible heat, Q=mLQ = mL for latent heat.
  • First Law: ΔU=Q−W\Delta U = Q - W — energy is conserved; work and heat are both forms of energy transfer.
  • Second Law sets a fundamental limit on engine efficiency: η≤1−TC/TH\eta \leq 1 - T_C/T_H (temperatures in kelvin).
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