Answer
58 g
Work Step by Step
The heat lost by the ice cube in cooling down to the temperature of liquid nitrogen goes into boiling the nitrogen. The nitrogen is already at its boiling point of 77K, or -196 degrees C.
$$Q=0=m_{ice}c_{ice}\Delta T+m_{N}L_N$$
$$-m_{ice}c_{ice}\Delta T=m_{N}L_N$$
$$-(0.028kg)(\frac{2100J}{kg\cdot C^{\circ}})(-196^{\circ}C-0^{\circ}C)=(m_N)(200\times10^3J/kg)$$
Solve for a nitrogen mass of 0.058 kg.
