An ideal monoatomic gas is enclosed in a rigid, perfectly spherical vessel at a pressure of 3 10^5 ~Pa . If the total internal energy of the gas is 4.8 10^6 ~J , the radius of the spherical vessel is _______ m .

Step-by-step solution

For an ideal monoatomic gas, the degree of freedom is f = 3 . The internal energy U of the gas can be expressed in terms of pressure P and volume V as: U = f 2 PV = 3 2 PV The volume of a spherical vessel of radius R is V = 4 3 R^3 . Substituting the volume into the internal energy equation: U = 3 2 P ( 4 3 R^3 ) = 2 P R^3 Given U = 4.8 10^6 ~J and P = 3 10^5 ~Pa , we have: 4.8 10^6 = 2 (3 10^5) R^3 4.8 10^6 = 6 10^5 R^3 48 = 6 R^3 R^3 = 8 R = 2 ~m Answer: 2