Новое решение

Valter правка от 5 авг. 2026 г., 17:46

правка #19851 позже →

en/5.5.1.md +39 −0

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		+	### Statement
		+
		+	$5.5.1.$ [Insert the problem statement]
		+
		+	### Solution
		+
		+	### Problem
		+
		+	$5.5.1.$ The volume of a gas was halved, and its temperature was increased by a factor of $1.5$. By what factor did the gas pressure increase?
		+
		+	### Solution
		+
		+	The amount of gas in this process remains constant, so for the two states (index 1 for the initial state, index 2 for the final state), we can write the ideal gas equation of state:
		+
		+	$$
		+	\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}
		+	$$
		+
		+	According to the problem statement, we are given: $V_2 = \frac{1}{2}V_1$ and $T_2 = \frac{3}{2}T_1$.
		+
		+	Substituting these into the equation:
		+
		+	$$
		+	\frac{P_1V_1}{T_1} = \frac{P_2 \cdot \frac{1}{2}V_1}{\frac{3}{2}T_1}
		+	$$
		+
		+	Hence:
		+
		+	$$
		+	P_2 = 3P_1
		+	$$
		+
		+	#### Answer:
		+
		+	By a factor of $3$
		+
		+	#### Answer
		+
		+	[Insert a concise answer or boxed result]

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			### Statement

			$5.5.1.$ [Insert the problem statement]

			### Solution

			### Problem

			$5.5.1.$ The volume of a gas was halved, and its temperature was increased by a factor of $1.5$. By what factor did the gas pressure increase?

			### Solution

			The amount of gas in this process remains constant, so for the two states (index 1 for the initial state, index 2 for the final state), we can write the ideal gas equation of state:

			$$
			\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}
			$$

			According to the problem statement, we are given: $V_2 = \frac{1}{2}V_1$ and $T_2 = \frac{3}{2}T_1$.

			Substituting these into the equation:

			$$
			\frac{P_1V_1}{T_1} = \frac{P_2 \cdot \frac{1}{2}V_1}{\frac{3}{2}T_1}
			$$

			Hence:

			$$
			P_2 = 3P_1
			$$

			#### Answer:

			By a factor of $3$

			#### Answer

			[Insert a concise answer or boxed result]