3. A 60 liter vessel is filled with oxygen at 125 bar Determine the final pressure of oxygen and the amount of heat supplied, if the initial temperature of oxygen is 10^circ C and the final tem- perature is 30^circ C Consider the heat capacity of oxygen as constant.

To solve this problem, we can use the ideal gas law and the first law of thermodynamics.<br /><br />Given information:<br />- Volume of the vessel: 60 liters<br />- Initial pressure of oxygen: 125 bar<br />- Initial temperature of oxygen: 10°C<br />- Final temperature of oxygen: 30°C<br />- Heat capacity of oxygen is constant<br /><br />Step 1: Convert the given temperatures to Kelvin.<br />Initial temperature (T1) = 10°C + 273.15 = 283.15 K<br />Final temperature (T2) = 30°C + 273.15 = 303.15 K<br /><br />Step 2: Use the ideal gas law to find the final pressure of oxygen.<br />The ideal gas law is given by the equation:<br />PV = nRT<br /><br />Where:<br />P = Pressure<br />V = Volume<br />n = Number of moles<br />R = Universal gas constant<br />T = Temperature<br /><br />Rearranging the equation, we get:<br />P = nRT / V<br /><br />Since the volume of the vessel is constant, the number of moles of oxygen remains the same. Therefore, we can write:<br />P1V1 = P2V2<br /><br />Substituting the given values, we get:<br />125 bar × 60 L = P260 L<br /><br />Solving for P2, we get:<br />P2 = 125 bar<br /><br />Therefore, the final pressure of oxygen is 125 bar.<br /><br />Step 3: Calculate the amount of heat supplied.<br />The first law of thermodynamics states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system.<br /><br />The change in internal energy (ΔU) can be calculated using the formula:<br />ΔU = nCpΔT<br /><br />Where:<br />n = Number of moles<br />Cp = Molar heat capacity at constant pressure<br />ΔT = Change in temperature<br /><br />Since the heat capacity of oxygen is constant, we can write:<br />ΔU = nCp(T2 - T1)<br /><br />Substituting the given values, we get:<br />ΔU = nCp(303.15 K - 283.15 K)<br /><br />Simplifying, we get:<br />ΔU = nCp(20 K)<br /><br />Therefore, the amount of heat supplied is nCp(20 K), where n is the number of moles of oxygen.