Problem Statement
An ideal gas expands freely (into vacuum) and doubles its volume at room temperature. Find $Q$, $W$, and $\Delta U$.
Given Information
- All quantities, constants, and constraints stated in the problem above
- Physical constants used as needed (see Concepts section)
Physical Concepts & Formulas
This problem draws on fundamental physical principles. The key is to identify which conservation law or field equation governs the system, then apply it systematically. Dimensional analysis can always be used to verify that the final answer has the correct units. Working from first principles — rather than memorising formulas — builds deeper understanding and allows tackling novel problems.
- Identify the relevant physical law (Newton’s laws, conservation of energy/momentum, Maxwell’s equations, etc.)
- State the mathematical form of that law as it applies here
- Check dimensions at every step: both sides of an equation must have the same units
Step-by-Step Solution
Problem Statement
An ideal gas expands freely (into vacuum) and doubles its volume at room temperature. Find $Q$, $W$, and $\Delta U$.
Given Information
- All quantities, constants, and constraints stated in the problem above
- Physical constants used as needed (see Concepts section)
Physical Concepts & Formulas
This problem draws on fundamental physical principles. The key is to identify which conservation law or field equation governs the system, then apply it systematically. Dimensional analysis can always be used to verify that the final answer has the correct units. Working from first principles — rather than memorising formulas — builds deeper understanding and allows tackling novel problems.
- Identify the relevant physical law (Newton’s laws, conservation of energy/momentum, Maxwell’s equations, etc.)
- State the mathematical form of that law as it applies here
- Check dimensions at every step: both sides of an equation must have the same units
Step-by-Step Solution
Problem Statement
An ideal gas expands freely (into vacuum) and doubles its volume at room temperature. Find $Q$, $W$, and $\Delta U$.
Given Information
- Free expansion into vacuum: no external pressure
- $V_2 = 2V_1$
- Ideal gas (internal energy depends only on T)
Physical Concepts & Formulas
In free expansion:
- $W = 0$ (no external pressure to push against, $P_{ext} = 0$)
- $Q = 0$ (process is too fast for heat exchange, or walls are adiabatic)
- From First Law: $\Delta U = Q – W = 0$
Step-by-Step Solution
Step 1: Work done.
$$W = \int P_{ext} dV = 0 \quad (P_{ext} = 0 \text{ for vacuum})$$
Step 2: Heat exchanged.
$$Q = 0 \quad (\text{adiabatic walls or too rapid for heat exchange})$$
Step 3: Change in internal energy (First Law).
$$\Delta U = Q – W = 0 – 0 = 0$$
Since $\Delta U = 0$ and the gas is ideal, $\Delta T = 0$ (temperature unchanged).
Worked Calculation
$$Q = 0, \quad W = 0, \quad \Delta U = 0$$
Answer
$Q = W = \Delta U = 0$; Temperature does not change.
Physical Interpretation
This is Joule’s experiment. For an ideal gas, free expansion is temperature-neutral. For real gases, molecules interact, so expanding against intermolecular attractions lowers temperature (Joule-Thomson cooling). This principle is used in refrigeration: Freon or CO₂ is throttled through a valve (essentially free expansion), dropping temperature and enabling cooling. The Joule-Thomson coefficient determines whether a gas heats or cools on expansion.
Worked Calculation
Substituting all given numerical values with their units into the derived formula:
$$\text{Numerical result} = \text{given expression substituted with values}$$
Answer
$$\boxed{Q = 0, \quad W = 0, \quad \Delta U = 0}$$
Physical Interpretation
The answer should be checked for dimensional consistency and physical reasonableness: is the magnitude in the expected range for this type of problem? Does the answer change in the correct direction when parameters are varied (e.g., increasing mass should increase momentum, increasing distance should decrease field strength)? These sanity checks are as important as the calculation itself.
Worked Calculation
Substituting all given numerical values with their units into the derived formula:
$$\text{Numerical result} = \text{given expression substituted with values}$$
Answer
$$\boxed{\boxed{Q = 0, \quad W = 0, \quad \Delta U = 0}}$$
Physical Interpretation
The answer should be checked for dimensional consistency and physical reasonableness: is the magnitude in the expected range for this type of problem? Does the answer change in the correct direction when parameters are varied (e.g., increasing mass should increase momentum, increasing distance should decrease field strength)? These sanity checks are as important as the calculation itself.
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