Book edition 1st Edition

Author(s) Paul Peter Urone

Pages 1272 pages

ISBN 9781938168000

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Short Answer

A bicycle tire has a pressure of 7.00 x 10⁵ N/m² at a temperature of $18.0 ° C$ and contains 2.00L of gas. What will its pressure be if you let out an amount of air that has a volume of 100 cm³ at atmospheric pressure? Assume tire temperature and volume remain constant.

The pressure of the gas becomes 6.93 x 10⁵ N/m².

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: State the ideal gas law

According to the ideal gas law,

PV = NkT

Here, P is the pressure of the gas, V is the volume occupied by the gas, N is the number of molecules in the gas, k is the Boltzmann constant, and T is the absolute temperature.

Step 2: Calculate the number of molecules present in the gas initially

Let P₀ be the initial pressure, N₀ be the number of molecules present initially, P_f be the final pressure, and N_f be the number of molecules at final pressure. Since volume and temperature are constant, the initial and final temperatures can be denoted as T Likewise, initial and final volume can be represented as V.

$\begin{array}{rcl} P_{0} & = & 7.00 \times 10^{5} N / m^{2} \\ T (K) & = & 18.0 ° C + 273.15 \\ T & = & 291.15 K \\ k & = & 1.38 \times 10^{- 23} J / K \end{array}$

$\begin{array}{rcl} V & = & \frac{2.00 L}{1000 L / m^{3}} \\ = & 2 \times 10^{- 3} m^{3} \end{array}$

Substitute the values in the equation

$\begin{array}{rcl} {7.00×10}^{5} N / m^{2} \times 2.00 \times 10^{- 3} m^{3} & = & N_{i} \times 1.38 \times 10^{- 23} J / K \times 291.15 K \\ N_{i} & = & \frac{7.00 \times 10^{5} N / m^{2} \times 2.00 \times 10^{- 3} m^{3}}{1.38 \times 10^{- 23} J / K \times 291.15 K} \\ N_{i} & = & 3.48 \times 10^{23} \end{array}$

Therefore, the number of molecules present in the gas initially is 3.48 x 10²³.

Step 3: Calculate the number of molecules removed from the tire

Let the volume of the air removed from the air be V₁ and its pressure will be equal to the atmospheric pressure. It can be denoted as P₁ . The number of molecules removed from the tire can be denoted as N₁.

$\begin{array}{rcl} P_{1} & = & 1.013 \times 10^{5} N / m^{2} \\ V_{1} & = & \frac{100 {cm}^{3}}{10^{6} {cm}^{3} / m^{3}} \\ = & 100 \times 10^{- 6} m^{3} \end{array}$

Substitute these values in the ideal gas equation.

$\begin{array}{rcl} {1.013×10}^{5} N / m^{2} \times 100 \times 10^{- 6} m^{3} & = & N_{1} \times 1.38 \times 10^{- 23} J / K \times 291.15 K \\ N_{i} & = & \frac{1.013 \times 10^{5} N / m^{2} \times 100 \times 10^{- 6} m^{3}}{1.38 \times 10^{- 23} J / K \times 291.15 K} \\ N_{i} & = & 2.52 \times 10^{21} \end{array}$

Therefore, the number of molecules removed from the tire is 2.52 x 10²¹.

Step 4: Calculate the final pressure

Since volume and temperature are constant, the unknown quantity that remains to calculate the pressure is the number of molecules. The number of molecules present at this stage is the difference between the number of molecules present initially and the number of molecules removed from the tire.

$\begin{array}{rcl} N_{f} & = & N_{0} - N_{1} \\ = & 3.48 \times 10^{23} - 2.52 \times 10^{21} \\ = & 3.45 \times 10^{23} \end{array}$

Substitute these values in the ideal gas equation.

$\begin{array}{rcl} P_{f} \times 2 \times 10^{- 3} m^{3} & = & 3.45 \times 10^{23} \times 1.38 \times 10^{- 23} J / K \times 291.15 K \\ P_{f} & = & \frac{3.45 \times 10^{23} \times 1.38 \times 10^{- 23} J / K \times 291.15 K}{2 \times 10^{- 3} m^{3}} \\ P_{f} & = & 6.93 \times 10^{5} N / m^{2} \end{array}$

Therefore, the final pressure is 6.93 x 10⁵ N/m².

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College Physics (Urone)

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Short Answer

A bicycle tire has a pressure of 7.00 x 10⁵ N/m² at a temperature of $18.0 ° C$ and contains 2.00L of gas. What will its pressure be if you let out an amount of air that has a volume of 100 cm³ at atmospheric pressure? Assume tire temperature and volume remain constant.

Step by Step Solution

Step 1: State the ideal gas law

Step 2: Calculate the number of molecules present in the gas initially

Step 3: Calculate the number of molecules removed from the tire

Step 4: Calculate the final pressure

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College Physics (Urone)

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A bicycle tire has a pressure of 7.00 x 105 N/m2 at a temperature of 18.0°C and contains 2.00L of gas. What will its pressure be if you let out an amount of air that has a volume of 100 cm3 at atmospheric pressure? Assume tire temperature and volume remain constant.

Step by Step Solution

Step 1: State the ideal gas law

Step 2: Calculate the number of molecules present in the gas initially

Step 3: Calculate the number of molecules removed from the tire

Step 4: Calculate the final pressure

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A bicycle tire has a pressure of 7.00 x 10⁵ N/m² at a temperature of $18.0 ° C$ and contains 2.00L of gas. What will its pressure be if you let out an amount of air that has a volume of 100 cm³ at atmospheric pressure? Assume tire temperature and volume remain constant.