Problem 4

Tennis balls are pressurised above atmospheric pressure for a good bounce; old hollow balls simply contained air at atmospheric pressure. Assume a ball inner radius

R = 3.0

cm constant during pressurisation, air composed of 20 % O2 and 80 % N2 by volume, and that overpressure does not expand the ball. (a) Calculate the mass

m

of air inside an old ball at

T_0 = 25

°C. (b) Modern balls are kept at

p_0 = 1.80

atm; premium balls contain pure N2. Once the can is opened, gas diffuses out until the inside reaches atmospheric pressure, with no inward diffusion; the process follows first-order kinetics in the overpressure. Premium balls depressurise to

p_1 = 1.40

atm after

t_1 = 241

h and to

p_2 = 1.19

atm after about

t_2 = 21

days at 25 °C. Demonstrate first-order behaviour and calculate the depressurisation rate constant

k_{\mathrm{N_2}}

in h

^{-1}

. (c) The initial depressurisation rate of regular (air) balls is 10 % faster than that of premium balls; calculate the rate constant

k_{\mathrm{O_2}}

assuming the rates of both gases are additive. (d) New premium balls, manufactured and canned at 25 °C, are opened and quickly brought to

T = 30.0

°C; a match starts

t = 12.0

h later. With

E_a = 50.0

kJ mol

^{-1}

for premium-ball depressurisation, calculate the ball pressure at the start of the match.

Step 3 of 4: Oxygen rate constant

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k_{\mathrm{air}} = 0.8\,k_{\mathrm{N_2}} + 0.2\,k_{\mathrm{O_2}} = 1.10\,k_{\mathrm{N_2}} \Rightarrow k_{\mathrm{O_2}} = 1.5\,k_{\mathrm{N_2}} = 4.3\times 10^{-3}\ \text{h}^{-1}

Analysis

For air (80 % N2, 20 % O2) the initial rates add: $v_{\text{air}} = k_{\mathrm{N_2}}\cdot 0.8\Delta p + k_{\mathrm{O_2}}\cdot 0.2\Delta p$ , while $v_{\text{N2}} = k_{\mathrm{N_2}}\Delta p$ . Setting $v_{\text{air}} = 1.10\,v_{\text{N2}}$ gives $0.8 + 0.2\,k_{\mathrm{O_2}}/k_{\mathrm{N_2}} = 1.10$ , so $k_{\mathrm{O_2}} = 1.5\,k_{\mathrm{N_2}} = 4.3\times 10^{-3}$ h $^{-1}$ (O2, being smaller... faster-diffusing, escapes more quickly than N2).