1. **State the problem:** We need to find the number of moles $n$ of an ideal gas sample given pressure $P$, volume $V$, and temperature $T$. 2. **Relevant formula:** The ideal gas law is $$n = \frac{PV}{RT}$$ where $P$ is pressure, $V$ is volume, $R$ is the gas constant, and $T$ is temperature in Kelvin. 3. **Given values:** - $P = 1.0 \times 10^5$ Pa - $V = 2.7 \times 10^{-3}$ m$^3$ - $T = 27^\circ C = 27 + 273 = 300$ K - $R = 8.31$ J/(mol·K) 4. **Calculate moles:** $$n = \frac{1.0 \times 10^5 \times 2.7 \times 10^{-3}}{8.31 \times 300}$$ 5. **Simplify numerator and denominator:** $$n = \frac{270}{2493}$$ 6. **Cancel common factors:** $$n = \frac{\cancel{270}}{\cancel{2493}} \approx 0.108$$ 7. **Interpretation:** The number of moles is approximately 0.11 mol. --- **Next problem:** Calculate the change in internal energy $\Delta U$ of a monatomic ideal gas when thermal energy $Q = 75$ J is added at constant pressure, volume changes from $V_1 = 2.7 \times 10^{-3}$ m$^3$ to $V_2 = 3.0 \times 10^{-3}$ m$^3$, and pressure $P = 1.0 \times 10^5$ Pa. 1. **Relevant formulas:** - Work done by gas: $$W = P \Delta V = P (V_2 - V_1)$$ - First law of thermodynamics: $$\Delta U = Q - W$$ 2. **Calculate work done:** $$W = 1.0 \times 10^5 \times (3.0 \times 10^{-3} - 2.7 \times 10^{-3}) = 1.0 \times 10^5 \times 0.3 \times 10^{-3} = 30 \text{ J}$$ 3. **Calculate change in internal energy:** $$\Delta U = 75 - 30 = 45 \text{ J}$$ 4. **Answer:** The change in internal energy is 45 J. **Final answers:** - Number of moles $n \approx 0.11$ mol - Change in internal energy $\Delta U = 45$ J