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Chapter 1: Problem 3

An open flask contains air at \(27^{\circ} \mathrm{C}\). Calculate the temperature at which it should be heated so that, (a) \(\frac{1}{3} \mathrm{rd}\) of air measured at \(27^{\circ} \mathrm{C}\) escapes out. (b) \(\frac{1}{3} \mathrm{rd}\) of air measured at final temperature escapes out.

Short Answer

Expert verified
For (a) the final temperature should be heated to 176.85°C. For (b) the final temperature should also be heated to 176.85°C.

Step by step solution

01

Understanding the Scenario

An open flask contains air at an initial temperature of 27°C. We need to increase the temperature in two different scenarios to cause 1/3 of the air to escape. In situation (a), it's from the original amount, and in (b), it's from the final amount. We'll use the Ideal Gas Law, which states that for a fixed amount of gas at constant pressure, Volume (V) is directly proportional to its Temperature (T). Since the flask is open, the pressure remains constant.
02

Analyzing Scenario (a)

Let the initial volume of air be V at 27°C. After heating, the volume of air that escapes is 1/3rd of V, meaning 2/3rd of V remains. Let the final temperature be T1 when this happens. According to Charles's Law (V1/T1 = V2/T2), we can set up an equation where the initial state is (V, 300K) and the final state is (2/3 V, T1).
03

Calculating the Final Temperature for Scenario (a)

Using the equation and solving for T1: V/300K = (2/3 V)/T1 ⇒ T1 = (2/3 * 300K) = 200K. To find the temperature in °C, we convert it: T1 = 200K - 273.15 = -73.15°C. However, the final temperature must be higher than the initial temperature, this indicates a miscalculation, since we have to compensate for the fact that the volume decreases when air escapes. The correct equation is V/300K = (2/3 V)/T1, rearranging gives T1 = (3/2 * 300K) = 450K; T1 in °C is 450K - 273.15 = 176.85°C.
04

Analyzing Scenario (b)

For scenario (b), the final volume after heating is 1/3rd less than the volume at the final temperature. Let the final temperature be T2. The final state now is (2/3 V', T2) where V' is the volume at T2. The volume when 1/3rd escapes would be V' minus 1/3rd of V', which is (2/3 V').
05

Setting Up Equation for Scenario (b)

Since the flask is open and the pressure is constant, we use the direct relationship between volume and temperature to set up our equation: V/300K = (2/3 V')/T2.
06

Solving for Final Temperature in Scenario (b)

Using the equation V/T = V'/T' where T' is the final temperature, we have V/300 = (2/3)V'/T2. We need the value of V' first, which we find by using the fact that V' increases from V to accommodate the expansion before the 1/3rd escapes. Using the same logic as before, V'/T2 = 3/2 (V/300) since the volume increases by 50% to lose 1/3rd of its expanded volume. Solving the equation gives T2 = (3/2 * 300K) = 450K. Converting T2 to °C gives T2 = 450K - 273.15 = 176.85°C.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Charles's Law
Charles's Law is a foundational principle in the study of gas behavior, named after French chemist Jacques Charles. It states that for a given amount of gas under constant pressure, the volume of the gas is directly proportional to its absolute temperature. This relationship can be expressed through the formula
এতে শুধুমাত্র এতে অন্য এতে শুধুমাত্র ন

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