The Eliza was built in Calcutta in 1806. This was her second voyage bringing convicts to New South Wales, the first being in 1820.
Surgeon William Rae
This was William Rae's first voyage as Surgeon Superintendent on a convict ship. He received a warrant of appointment as Surgeon Superintendent on 19th June 1822 and proceeded to Deptford that same day to join the ship where the Guard had already embarked.
William Rae was also employed as surgeon on convict ships:
On 30th June the ship sailed for Sheerness where 50 convicts were received from the Ganymede Hulk on 2nd July. The following morning another 55 convicts came on board from the Bellerophon and the same number from the Retribution hulk at Sheerness. Sixteen boys were allotted a separate prison. On 11 July, the surgeon recorded that all the men were allowed on deck during the day when they were frequently visited by their friends and relations. As most of them only embarked with the clothes they stood in, they were supplied with a shirt and pair of trousers each. 160 shirts and trousers were issued. 
On the 16th July a packet and a bag of despatches for the Governor of New South Wales and a despatch to the master of the ship ordering the Eliza to proceed on her voyage to New South Wales were received on board. Three days later the convicts were all on deck taking a last farewell of their friends and relations. A few seemed to feel the situation deeply but the majority according to the surgeon appear to be callous and behave with that stoicism and indifference which can only be found amongst men inured to villainy and hardened with vice.
The following morning, 20th July 1822, they weighed anchor and sailed for the Downs which they came to anchor at dusk. Most of the convicts and passengers were sea sick.
Bibles, testaments and prayer books were distributed amongst the convicts and also a few books and writing implements from the surgeon's own store were given to the boys who soon made considerable improvement in their learning. The youngest prisoners were Thomas Ball (16); Murdock Chisholm (16); Benjamin Johnson (16); William McCoy (16); William McNicholl (16); William Redgate (15); James Statham (16); Matthew Sullivan(15); George Williams (14); and Joseph Windle (16).
A week after departing the Eliza struck bad weather. There were strong gales with rain from the SW with the ship pitching frighteningly and they were obliged to anchor in Dungeness.
They reached the equator on 10 September. The Convicts were all on deck during the morning, but afterwards ordered below until the sailors and soldiers had performed the usual ceremony at crossing the equator. The prisoners, however were all very merry amongst themselves and during their temporary confinement did not let the said ceremony pass unobserved. They constituted barbers and with a little suet and shoe blacking and a bullocks rib for a razor shaved every individual in the prison. All submitting to the operation with much good humour. 
On the evening of the 19 October 1822 several of the prisoners, (amateurs) in testimony of the gratitude which they felt for the liberty they had hitherto enjoyed and the various indulgences which had been granted to them since their embarkation, entertained the officers with the performance of the play Rob Roy.
They sailed close by the island of St. Pauls on 25 October.
The Eliza reached Port Jackson on 22 November 1822.
One hundred and sixty male prisoners were landed in good health on 26th November 1822. They had been on board for 147 days and the voyage had taken 125 days.
After landing, the convicts were assigned to various settlers and public works at Windsor, Upper Minto, Airds, Penrith, Emu Plains and Bathurst.
Surgeon William Rae was given an allowance of £50 for the return voyage to England. He departed on the Castle Forbes for Van Diemen's Land in December 1822 and may have continued to England on this vessel.
The Eliza departed for Batavia in January 1823. 1st Officer Mr. Hustwick; 2nd Officer Mr. Faith; 3rd Officer Mr. Robinson. Passengers John Spain, Joseph Hall, Richard Rexworthy and Joseph Brown.
MR. FRANCIS MITCHELL. This gentleman expired at his residence, Darlinghurst Road at 8 o'clock on Monday morning, at the advanced age of seventy-two, after a short illness. In the death of Mr. Francis Mitchell we have lost one of those early settlers of the colony, of more than half a century, that link past associations with the present time. Upwards of fifty-five years ago Mr. Mitchell came to this colony, and was for many years associated with the late Alexander Berry and Mr. Wolstencroft in business and subsequently for thirty-five years he occupied the position of senior partner of the firm of Messrs. Mitchell and Co. but some sixteen years back he retired from business, having acquired a sufficient competence, and since then his time has been principally devoted to matters of a charitable nature. For many years he was a director of the Bank of New South Wales, and up to within a few days of his death he attended to his duties as trustee of the Saving Bank, which institution lie has been connected with for more than twenty years. Personally, Mr. Mitchell was of a retiring disposition, and free from all ostentation ; while philanthropic and charitable, lie always tempered his benevolence with prudence, and in his death the colony has lost an old and valued friend, whose many acts of kindness have won for him the esteem and respect of a wide circle of friends and acquaintances. - Sydney Mail 22 July 1876
Alexander Cunningham, Robert Robertson, and Jean Robertson, stood indicted for eleven acts of house. breaking, theft, and reset of theft, to nine of which charges Cunningham pleaded guilty, but denied being habit and repute a thief. This prisoner is only eighteen years of age. Robert Robertson, only fourteen years of age, pleaded guilty to the second, third, and fourth charges; Jane Robertson at first pleaded not guilty, but retracted her plea, and acknowledged being guilty of the ninth charge, and habit and repute a thief. A Jury having been chosen, the prisoners were found guilty; the Jury recommending Cunningham to mercy, on account of his youth. He was sentenced to be executed on the 27th February, and the Robertsons to be transported for fourteen years. (Cunningham has been since respited.)
3). Prisoners mentioned in the Surgeon's Journal:
Selected 16 boys for the prison allotted to them and appointed John White and Jeremy Garfield to superintend them.
4 July 1822: Appointed Joseph Morbee and James Edrop to be captains of the prison deck, and Labian White and John Matthewson to attend the sick and wait in the hospital.
15 July 1822: Convicts all upon deck and this morning notwithstanding the many faults and transgressions which had been looked over, two of them, Michael Garain and Edward Gardiner commenced a regular pugilistic match upon deck; as it were in open defiance of all authority, being determined to check every growing evil in its bud, therefore, punished the offenders with one dozen of lashes each. Wherefore they expressed sincere contrition tho previously they were not willing to acknowledge themselves guilty of any offence. Sick five.
16 July 1822: This morning Joseph Bates was accused by Richard Childs of having stolen from him during the night six pounds of sugar and a quantity of pipe clay. Made search for said articles and found the sugar in Bate's bed and part of the pipe clay in his pockets. As this crime is likely to be such a cause of evil and disturbance during the voyage, therefore, punished the offender with one dozen and a half lashes and held him out as an example to others if found guilty of a similar offence.
26 July 1822: Distributed bibles, testaments, prayer books and psalters and manuals of devotion amongst them. A few days ago I also served out a few books which were furnished to me privately amongst the boys at the same time furnished them with materials for writing and I am happy to observe that they are already making considerable improvement under their teachers. Sick, nine.
30 July 1822: Last night considerable noise in the afterpart of the prison and the centinel called out that they were breaking open boxes. Opened the prison and found a box belonging to Richard Naylor forced open; but fortunately the intended thief had not time to remove anything from it as the captains of the deck sprung from their bed upon the alarm being given and prevented any one from passing forward until a light was procured when George Marshall was found in that part of the Prison dressed and the only person out of bed. Strong suspicion, therefore, attached itself to him and the noise in the after Berths left no doubt that he had accomplices in the intended robbing. This morning called the parties upon deck informed them of the suspicion attached to them. That their conduct would be narrowly observed in future and that but for the want of sufficient proof on this occasion they would have been punished in the most exemplary manner.
15 August 1822: Confined John Connor on bread and water until the evening for being insolent to the Captain on deck.
16 August 1822: Confined J P Lawsor on bread and water until the evening for being insolent to one of the officers. Sick, nine.
23 August 1822: Confined John Carr on bread and water until the evening for insolence and bad language to one of his comrades. Sick, seven.
1). Bateson, Charles Library of Australian History (1983). The convict ships, 1787-1868 (Australian ed). Library of Australian History, Sydney : pp.344-345, 384
2). National Archives - Reference: ADM 101/23/3/1 Transcript of the Medical and surgical journal of the convict ship Eliza by surgeon William Rae
3). Ancestry.com. UK, Royal Navy Medical Journals, 1817-1857. Medical Journal of William Rae on the voyage of the Eliza in 1822. The National Archives. Kew, Richmond, Surrey.