The drawing shows an ideal gas confined to a cylinder by a massless piston that is attached to an ideal spring. Outside the cylinder is a vacuum. The cross-sectional area of the piston is \( A = 2.50 \times 10^{-3} \, \text{m}^2 \). The initial pressure, volume, and temperature of the gas are, respectively, \( P_0, V_0 = 6.00 \times 10^{-4} \, \text{m}^3 \) and \( T_0 = 273 \, \text{K} \), and the spring is initially stretched by an amount \( x_0 = 0.096 \, \text{m} \) with respect to its unstrained length. The gas is heated, so that its final pressure, volume, and temperature are \( P_f, V_f \) and \( T_f \) and the spring is stretched by an amount \( x_f = 0.14 \, \text{m} \) with respect to its unstrained length. What is the final temperature of the gas?Below the text is a box for the final temperature, labeled \( T_f = \), designed for user input.The image includes a diagram of a cylindrical container with a piston on top. The piston is depicted as a flat, horizontal disk inside the cylinder, and an arrow indicates the direction against which it is fighting the spring. The spring is shown compressing or expanding in response to pressure changes inside the cylinder. The visual emphasizes the mechanical setup and the interaction between the gas and the spring-loaded piston.

In initial pressure is P0 and the elongation of spring is x0, then P0A=kx0 Where P0A is the force on the piston and k is the spring constant. Similarly if final pressure is Pf and elongation of spring is xf, then PfA=kxf Dividing the two equations we get P0APfA=kx0kxf⇒P0Pf=x0xf⇒Pf=xfx0P0 ---(1)Now using the gas law P0V0T0=PfVfTf⇒Tf=PfVfP0V0T0 We continue further after using equation (1). Tf=PfVfP0V0T0=xfx0P0VfP0V0T0=xfVfx0V0T0---(2)Now let the natural length of spring be l. Then we have for volume of piston V0=l+x0A⇒l=V0A-x0 and for final volume, Vf=l+xfA=V0A-x0+xfA Substitute this in equation (2) to get, Tf=xfVfx0V0T0=xfAV0A-x0+xfx0V0T0=xfx01+xf-x0AV0T0=0.140.0961+0.14-0.0962.5·10-36·10-4·273=471.1145833 K

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Physics

The drawing shows an ideal gas confined to a cylinder by a massless piston that is attached to an ideal spring. Outside the cylinder is a vacuum. The cross-sectional area of the piston is A = 2.50 x 103 m². The initial pressure, volume, and temperature of the gas are, respectively, Po, Vo = 6.00 x 104 m³ and To = 273 K, and the spring is initially stretched by an amount xo = 0.096 m with respect to its unstrained length. The gas is heated, so that its final pressure, volume, and temperature are P₁, Vf and T, and the spring is stretched by an amount x₁ = 0.14 m with respect to its unstrained length. What is the final temperature of the gas? T₁ = i Piston

The drawing shows an ideal gas confined to a cylinder by a massless piston that is attached to an ideal spring. Outside the cylinder is a vacuum. The cross-sectional area of the piston is A = 2.50 x 103 m². The initial pressure, volume, and temperature of the gas are, respectively, Po, Vo = 6.00 x 104 m³ and To = 273 K, and the spring is initially stretched by an amount xo = 0.096 m with respect to its unstrained length. The gas is heated, so that its final pressure, volume, and temperature are P₁, Vf and T, and the spring is stretched by an amount x₁ = 0.14 m with respect to its unstrained length. What is the final temperature of the gas? T₁ = i Piston

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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